Package 'MOTE'

Format numbers for APA-style reporting

Description

Create "pretty" character representations of numeric values with a fixed number of decimal places, optionally keeping or omitting the leading zero for values between -1 and 1.

Usage

apa(value, decimals = 3, leading = TRUE)

Arguments

value	Numeric input: a single number, vector, matrix, or a data frame with all-numeric columns. Non-numeric inputs will error.
decimals	A single non-negative integer giving the number of decimal places to keep in the output.
leading	Logical: 'TRUE' to keep leading zeros on decimals (e.g., '0.25'), 'FALSE' to drop them (e.g., '.25'). Default is 'TRUE'.

Details

This function formats numbers for inclusion in manuscripts and reports. - When 'leading = TRUE', numbers are rounded and padded to 'decimals' places, keeping the leading zero for values with absolute value < 1. - When 'leading = FALSE', the leading zero before the decimal point is removed for values with absolute value < 1. If 'value' is a data frame, all columns must be numeric; otherwise an error is thrown.

Value

A character vector/array (matching the shape of 'value') containing the formatted numbers.

Examples

apa(0.54674, decimals = 3, leading = TRUE)   # "0.547"
apa(c(0.2, 1.2345, -0.04), decimals = 2)     # "0.20" "1.23" "-0.04"
apa(matrix(c(0.12, -0.9, 2.3, 10.5), 2), decimals = 1, leading = FALSE)
# returns a character matrix with ".1", "-.9", "2.3", "10.5"

---

Between-Subjects One-Way ANOVA Example Data

Description

Ratings of close interpersonal attachments for 45-year-old participants, categorized by self-reported health status: excellent, fair, or poor. This dataset is designed for use with functions such as eta.F, eta.full.SS, omega.F, omega.full.SS, and epsilon.full.SS.

Usage

data(bn1_data)

Format

A data frame with *n* rows and 2 variables:

group

Factor with levels "poor", "fair", and "excellent".

friends

Numeric rating of close interpersonal attachments.

Source

Simulated data inspired by Nolan & Heinzen (4th ed.), *Statistics for the Behavioral Sciences*. Generated for instructional examples in the MOTE package.

References

Nolan, S. A., & Heinzen, T. E. (*4th ed.*). *Statistics for the Behavioral Sciences*. Macmillan Learning.

---

Between-Subjects Two-Way ANOVA Example Data

Description

Example data for a between-subjects two-way ANOVA examining whether athletic spending differs by sport type and coach experience. This dataset contains simulated athletic budgets (in thousands of dollars) for baseball, basketball, football, soccer, and volleyball teams, with either a new or old coach. Designed for use with omega.partial.SS.bn, eta.partial.SS, and other between-subjects ANOVA designs.

Usage

data(bn2_data)

Format

A data frame with 3 variables:

coach

Factor with levels "old" and "new" indicating coach experience.

type

Factor indicating sport type: "baseball", "basketball", "football", "soccer", or "volleyball".

money

Numeric. Athletic spending in thousands of dollars.

Source

Simulated data generated for instructional examples in the MOTE package.

---

Chi-Square Test Example Data

Description

Example data for a chi-square test of independence. Individuals were polled and asked to report their number of friends (low, medium, high) and their number of children (1, 2, 3 or more). The analysis examines whether there is an association between friend group size and number of children. It was hypothesized that those with more children may have less time for friendship-maintaining activities.

Usage

data(chisq_data)

Format

A data frame with 2 variables:

friends

Factor with levels "low", "medium", and "high" indicating self-reported number of friends.

kids

Factor with levels "1", "2", and "3+" indicating number of children.

Source

Simulated data inspired by Nolan & Heinzen (4th ed.), *Statistics for the Behavioral Sciences*. Generated for instructional examples in the MOTE package.

References

Nolan, S. A., & Heinzen, T. E. (*4th ed.*). *Statistics for the Behavioral Sciences*. Macmillan Learning.

---

Confidence interval for R^2 (exported helper)

Description

Compute a confidence interval for the coefficient of determination (R^2). This implementation follows MBESS (Ken Kelley) and is exported here to avoid importing many dependencies. It supports cases with random or fixed predictors and can be parameterized via either degrees of freedom or sample size (n) and number of predictors (p/k).

Usage

ci_r2(
  r2 = NULL,
  df1 = NULL,
  df2 = NULL,
  conf_level = 0.95,
  random_predictors = TRUE,
  random_regressors = random_predictors,
  f_value = NULL,
  n = NULL,
  p = NULL,
  k = NULL,
  alpha_lower = NULL,
  alpha_upper = NULL,
  tol = 1e-09
)

Arguments

r2	Numeric. The observed R^2 (may be 'NULL' if 'f_value' is supplied).
df1	Integer. Numerator degrees of freedom from F.
df2	Integer. Denominator degrees of freedom from F.
conf_level	Numeric in (0, 1). Two-sided confidence level for a symmetric confidence interval. Default is '0.95'. Cannot be used with 'alpha_lower' or 'alpha_upper'.
random_predictors	Logical. If 'TRUE' (default), compute limits for random predictors; if 'FALSE', compute limits for fixed predictors.
random_regressors	Logical. Backwards-compatible alias for 'random_predictors'. If supplied, it overrides 'random_predictors'.
f_value	Numeric. The observed F statistic from the study.
n	Integer. Sample size.
p	Integer. Number of predictors.
k	Integer. Alias for 'p' (number of predictors). If supplied along with 'p', they must be equal.
alpha_lower	Numeric. Lower-tail noncoverage probability (cannot be used with 'conf_level').
alpha_upper	Numeric. Upper-tail noncoverage probability (cannot be used with 'conf_level').
tol	Numeric. Tolerance for the iterative method determining critical values. Default is '1e-9'.

Details

If 'n' and 'p' (or 'k') are provided, 'df1' and 'df2' are derived as 'df1 = p' and 'df2 = n - p - 1'. Conversely, if 'df1' and 'df2' are provided, 'n = df1 + df2 + 1' and 'p = df1'.

Value

A named list with the following elements:

lower_conf_limit_r2

The lower confidence limit for R^2.

prob_less_lower

Probability associated with values less than the lower limit.

upper_conf_limit_r2

The upper confidence limit for R^2.

prob_greater_upper

Probability associated with values greater than the upper limit.

References

Kelley, K. (2007). Methods for the behavioral, educational, and social sciences: An R package (MBESS).

---

Cohen's d for Paired t Using the Average SD Denominator

Description

**Note on function names:** This function now uses the snake_case name 'd_dep_t_avg()' to follow modern R style guidelines and CRAN recommendations. The dotted version 'd.dep.t.avg()' is still included as a wrapper for backward compatibility, so older code will continue to work. Both functions produce identical results, but new code should use 'd_dep_t_avg()'. The output function also provides backwards compatibility and new snake case variable names.

Usage

d_dep_t_avg(m1, m2, sd1, sd2, n, a = 0.05)

d.dep.t.avg(m1, m2, sd1, sd2, n, a = 0.05)

Arguments

m1	Mean from the first level/occasion.
m2	Mean from the second level/occasion.
sd1	Standard deviation from the first level/occasion.
sd2	Standard deviation from the second level/occasion.
n	Sample size (number of paired observations).
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

Compute Cohen's $d_{av}$ and a noncentral-t confidence interval for repeated-measures (paired-samples) designs using the **average of the two standard deviations** as the denominator.

The effect size is defined as the mean difference divided by the average SD:

$d_{av} = \frac{m_1 - m_2}{\left( s_1 + s_2 \right)/2}.$

The test statistic used for the noncentral-t confidence interval is based on the average of the two standard errors, $se_i = s_i/\sqrt{n}$:

$t = \frac{m_1 - m_2}{\left( \frac{s_1}{\sqrt{n}} + \frac{s_2}{\sqrt{n}} \right) / 2}.$

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d_{av}$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d_{av}$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d_{av}$.

M1, M2

Group means.

M1low, M1high, M2low, M2high

Confidence interval bounds for each mean.

sd1, sd2

Standard deviations.

se1, se2

Standard errors of the means.

n

Sample size.

df

Degrees of freedom ($n - 1$).

estimate

APA-style formatted string for reporting $d_{av}$ and its CI.

Examples

# The following example is derived from the "dept_data" dataset included
# in the MOTE package.

# Suppose seven people completed a measure of belief in the supernatural
# before and after watching a sci-fi movie.
# Higher scores indicate stronger belief.

    t.test(dept_data$before, dept_data$after, paired = TRUE)

# You can type in the numbers directly, or refer to the
# dataset, as shown below.

    d_dep_t_avg(m1 = 5.57, m2 = 4.43, sd1 = 1.99,
                sd2 = 2.88, n = 7, a = .05)

    d_dep_t_avg(5.57, 4.43, 1.99, 2.88, 7, .05)

    d_dep_t_avg(mean(dept_data$before), mean(dept_data$after),
                sd(dept_data$before), sd(dept_data$after),
                length(dept_data$before), .05)

---

Cohen's d for Paired t Using the SD of Difference Scores

Description

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'd_dep_t_diff()' to follow modern R style guidelines. The original dotted version 'd.dep.t.diff()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'mdiff', 'Mlow', 'Mhigh', 'sddiff') and newer snake_case aliases (e.g., 'm_diff', 'm_diff_lower_limit', 'm_diff_upper_limit', 'sd_diff'). New code should prefer 'd_dep_t_diff()' and the snake_case output names, but existing code using the older names will continue to work.

Usage

d_dep_t_diff(mdiff, sddiff, n, a = 0.05)

d.dep.t.diff(mdiff, sddiff, n, a = 0.05)

Arguments

mdiff	Mean of the difference scores.
sddiff	Standard deviation of the difference scores.
n	Sample size (number of paired observations).
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

Compute Cohen's $d_z$ and a noncentral-t confidence interval for repeated-measures (paired-samples) designs using the **standard deviation of the difference scores** as the denominator.

The effect size is defined as:

$d_z = \frac{\bar{X}_D}{s_D}$

where $\bar{X}_D$ is the mean of the difference scores and $s_D$ is the standard deviation of the difference scores.

The corresponding t statistic for the paired-samples t-test is:

$t = \frac{\bar{X}_D}{s_D / \sqrt{n}}$

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d_z$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d_z$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d_z$.

mdiff

Mean difference score.

Mlow, Mhigh

Confidence interval bounds for the mean difference.

sddiff

Standard deviation of the difference scores.

se

Standard error of the difference scores.

n

Sample size.

df

Degrees of freedom ($n - 1$).

t

t-statistic.

p

p-value.

estimate

APA-style formatted string for reporting $d_z$ and its CI.

statistic

APA-style formatted string for reporting the t-statistic and p-value.

Examples

# Example derived from the "dept_data" dataset included in MOTE

# Suppose seven people completed a measure of belief in the supernatural
# before and after watching a sci-fi movie.
# Higher scores indicate stronger belief.

t.test(dept_data$before, dept_data$after, paired = TRUE)

# Direct entry of summary statistics:
d_dep_t_diff(mdiff = 1.14, sddiff = 2.12, n = 7, a = .05)

# Equivalent shorthand:
d_dep_t_diff(1.14, 2.12, 7, .05)

# Using raw data from the dataset:
d_dep_t_diff(mdiff = mean(dept_data$before - dept_data$after),
             sddiff = sd(dept_data$before - dept_data$after),
             n = length(dept_data$before),
             a = .05)

---

Cohen's d from t for Paired Samples Using the SD of Difference Scores

Description

Compute Cohen's $d_z$ from a paired-samples t-statistic and provide a noncentral-t confidence interval, using the **standard deviation of the difference scores** as the denominator.

Usage

d_dep_t_diff_t(t_value, t = NULL, n, a = 0.05)

Arguments

t_value	t-statistic from a paired-samples t-test.
t	for backwards compatibility, you can also give t.
n	Sample size (number of paired observations).
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

For paired designs, $d_z$ can be obtained directly from the t-statistic:

$d_z = \frac{t}{\sqrt{n}},$

where $n$ is the number of paired observations (df = $n-1$). The $(1-\alpha)$ confidence interval for $d_z$ is derived from the noncentral t distribution for the observed $t$ and df.

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d_z$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d_z$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d_z$.

n

Sample size.

df

Degrees of freedom ($n - 1$).

t

t-statistic.

p

p-value.

estimate

APA-style formatted string for reporting $d_z$ and its CI.

statistic

APA-style formatted string for reporting the t-statistic and p-value.

Examples

# Example derived from the "dept_data" dataset included in MOTE

# Suppose seven people completed a measure before and after an intervention.
# Higher scores indicate stronger endorsement.

    scifi <- t.test(dept_data$before, dept_data$after, paired = TRUE)

# The t-test value was 1.43. You can type in the numbers directly,
# or refer to the dataset, as shown below.

    d_dep_t_diff_t(t_value = 1.43, n = 7, a = .05)

    d_dep_t_diff_t(t_value = scifi$statistic,
        n = length(dept_data$before), a = .05)

---

Cohen's d for Paired t Controlling for Correlation (Repeated Measures)

Description

Compute Cohen's $d_{rm}$ and a noncentral-t confidence interval for repeated-measures (paired-samples) designs **controlling for the correlation between occasions**. The denominator uses the SDs and their correlation.

Usage

d_dep_t_rm(m1, m2, sd1, sd2, r, n, a = 0.05)

Arguments

m1	Mean from the first level/occasion.
m2	Mean from the second level/occasion.
sd1	Standard deviation from the first level/occasion.
sd2	Standard deviation from the second level/occasion.
r	Correlation between the two levels/occasions.
n	Sample size (number of paired observations).
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

The effect size is defined as:

$d_{rm} = \frac{m_1 - m_2}{\sqrt{s_1^2 + s_2^2 - 2 r s_1 s_2}} \; \sqrt{2(1-r)}.$

The test statistic used for the noncentral-t confidence interval is:

$t = \frac{m_1 - m_2}{\sqrt{\dfrac{s_1^2 + s_2^2 - 2 r s_1 s_2}{n}}} \; \sqrt{2(1-r)}.$

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d_{rm}$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d_{rm}$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d_{rm}$.

M1, M2

Group means.

M1low, M1high, M2low, M2high

Confidence interval bounds for each mean.

sd1, sd2

Standard deviations.

se1, se2

Standard errors of the means.

r

Correlation between occasions.

n

Sample size.

df

Degrees of freedom ($n - 1$).

estimate

APA-style formatted string for reporting $d_{rm}$ and its CI.

Examples

# Example derived from the "dept_data" dataset included in MOTE

    t.test(dept_data$before, dept_data$after, paired = TRUE)

    scifi_cor <- cor(dept_data$before, dept_data$after, method = "pearson",
                     use = "pairwise.complete.obs")

# Direct entry of summary statistics, or refer to the dataset as shown below.

    d_dep_t_rm(m1 = 5.57, m2 = 4.43, sd1 = 1.99,
               sd2 = 2.88, r = .68, n = 7, a = .05)

    d_dep_t_rm(5.57, 4.43, 1.99, 2.88, .68, 7, .05)

    d_dep_t_rm(mean(dept_data$before), mean(dept_data$after),
               sd(dept_data$before), sd(dept_data$after),
               scifi_cor, length(dept_data$before), .05)

---

General interface for Cohen's d

Description

'd_effect()' is a convenience wrapper that will route to the appropriate Cohen's *d* helper function based on the arguments supplied. This allows users to call a single function for different study designs while maintaining backward compatibility with the more specific helpers.

Usage

d_effect(
  m1 = NULL,
  m2 = NULL,
  sd1 = NULL,
  sd2 = NULL,
  u = NULL,
  sig = NULL,
  r = NULL,
  mdiff = NULL,
  sddiff = NULL,
  t_value = NULL,
  z_value = NULL,
  p1 = NULL,
  p2 = NULL,
  n1 = NULL,
  n2 = NULL,
  n = NULL,
  a = 0.05,
  design,
  ...
)

Arguments

m1	Means of the two conditions or measurements.
m2	Means of the two conditions or measurements.
sd1	Standard deviations for the two conditions or measurements.
sd2	Standard deviations for the two conditions or measurements.
u	Population or comparison mean for one‑sample t‑designs, used when 'design = "single_t"'.
sig	Population standard deviation for z-based designs, used when 'design = "z_mean"'.
r	Correlation between the paired measurements (used for repeated-measures designs such as '"dep_t_rm"').
mdiff	Mean difference between paired observations.
sddiff	Standard deviation of the difference scores.
t_value	t statistic value for the test. Used in designs where the effect size is derived directly from a reported t-value (e.g., '"dep_t_diff_t"', '"ind_t_t"', or '"single_t_t"').
z_value	z statistic value for the test. Used in designs where the effect size is derived directly from a reported z-value (e.g., '"z_z"').
p1	Proportion for group one (between 0 and 1), used in the '"prop"' design.
p2	Proportion for group two (between 0 and 1), used in the '"prop"' design.
n1	Sample sizes for the two independent groups (used for independent-groups designs such as '"ind_t"').
n2	Sample sizes for the two independent groups (used for independent-groups designs such as '"ind_t"').
n	Sample size (number of paired observations).
a	Significance level used when computing confidence intervals. Defaults to '0.05'.
design	Character string specifying the study design.
...	Reserved for future arguments and passed on to the underlying helper functions when appropriate.

Details

- '"delta_ind_t"' — independent-groups t-test using the delta effect size, where the SD of group 1 is used as the denominator. Supply 'm1', 'm2', 'sd1', 'sd2', 'n1', and 'n2'. In this case, 'd_effect()' will call [delta.ind.t()] with the same arguments.

- ‘"g_ind_t"' — independent-groups t-test using Hedges’ g, which applies a small-sample correction to the standardized mean difference. Supply 'm1', 'm2', 'sd1', 'sd2', 'n1', and 'n2'. In this case, 'd_effect()' will call [g_ind_t()] with the same arguments.

- '"z_z"' — one-sample z-test effect size where the *z* value is supplied directly along with the sample size 'n'. Supply 'z_value' and 'n'. You may optionally supply 'sig' (population SD) for descriptive reporting. In this case, 'd_effect()' will call [d_z_z()] with the same arguments.

Value

A list with the same structure as returned by the underlying helper function. For the current paired-means case, this is the output of [d_dep_t_avg()], which includes:

• ‘d' – Cohen’s d using the average SD denominator.

• 'dlow', 'dhigh' – lower and upper confidence limits for 'd'.

• Snake_case aliases such as 'd_lower_limit' and 'd_upper_limit'.

• Descriptive statistics (means, SDs, SEs, and their confidence limits) for each group.

Supported designs

- '"dep_t_avg"' — paired/dependent t-test with average SD denominator. Supply 'm1', 'm2', 'sd1', 'sd2', and 'n'. In this case, 'd()' will call [d_dep_t_avg()] with the same arguments.

- '"dep_t_diff"' — paired/dependent t-test using the **SD of the difference scores**. Supply 'mdiff', 'sddiff', and 'n'. In this case, 'd()' will call [d_dep_t_diff()] with the same arguments.

- '"dep_t_diff_t"' — paired/dependent t-test where the *t* value is supplied directly. Supply 't_value' and 'n'. In this case, 'd()' will call [d_dep_t_diff_t()] with the same arguments.

- '"dep_t_rm"' — paired/dependent t-test using the repeated-measures effect size $d_{rm}$, which adjusts for the correlation between measurements. Supply 'm1', 'm2', 'sd1', 'sd2', 'r', and 'n'. In this case, 'd()' will call [d_dep_t_rm()] with the same arguments.

- '"ind_t"' — independent-groups t-test using the pooled SD (\(d_s\)). Supply 'm1', 'm2', 'sd1', 'sd2', 'n1', and 'n2'. In this case, 'd()' will call [d_ind_t()] with the same arguments.

- '"ind_t_t"' — independent-groups t-test where the *t* value is supplied directly. Supply 't_value', 'n1', and 'n2'. In this case, 'd()' will call [d_ind_t_t()] with the same arguments.

- ‘"g_ind_t"' — independent-groups t-test using Hedges’ g, which applies a small-sample correction to the standardized mean difference. Supply 'm1', 'm2', 'sd1', 'sd2', 'n1', and 'n2'. In this case, 'd_effect()' will call [g_ind_t()] with the same arguments.

- '"single_t"' — one‑sample t‑test effect size using the sample mean, population mean, sample SD, and sample size. Supply 'm1' (sample mean), 'u' (population mean), 'sd1', and 'n'. In this case, 'd()' will call [d_single_t()] with the same arguments.

- '"single_t_t"' — one-sample t-test effect size where the *t* value is supplied directly along with the sample size 'n'. In this case, 'd()' will call [d_single_t_t()] with the same arguments.

- '"prop"' — independent proportions (binary outcome) using a standardized mean difference (SMD) that treats each proportion as the mean of a Bernoulli variable with pooled Bernoulli SD. Supply 'p1', 'p2', 'n1', and 'n2'. In this case, 'd()' will call [d_prop()] with the same arguments.

- ‘"prop_h"' — independent proportions (binary outcome) using Cohen’s \(h\) based on the arcsine-transformed difference between proportions. Supply 'p1', 'p2', 'n1', and 'n2'. In this case, 'd()' will call [h_prop()] with the same arguments.

- '"z_mean"' — one-sample z-test effect size using a known population standard deviation. Supply 'm1' (sample mean), 'u' (population mean), 'sd1' (sample SD, used for descriptive CIs), 'sig' (population SD), and 'n'. In this case, 'd_effect()' will call [d_z_mean()] with the same arguments.

Examples

# Paired/dependent t-test using average SD denominator
# These arguments will route d() to d_dep_t_avg()
d_effect(
  m1 = 5.57, m2 = 4.43,
  sd1 = 1.99, sd2 = 2.88,
  n = 7, a = .05,
  design = "dep_t_avg"
)

# You can also call the helper directly
d_dep_t_avg(
  m1 = 5.57, m2 = 4.43,
  sd1 = 1.99, sd2 = 2.88,
  n = 7, a = .05
)

---

Cohen's d for Independent Samples Using the Pooled SD

Description

Compute Cohen's $d_s$ for between-subjects designs and a noncentral-t confidence interval using the **pooled standard deviation** as the denominator.

Usage

d_ind_t(m1, m2, sd1, sd2, n1, n2, a = 0.05)

Arguments

m1	Mean of group one.
m2	Mean of group two.
sd1	Standard deviation of group one.
sd2	Standard deviation of group two.
n1	Sample size of group one.
n2	Sample size of group two.
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

The pooled standard deviation is:

$s_{pooled} = \sqrt{ \frac{ (n_1 - 1)s_1^2 + (n_2 - 1)s_2^2 } {n_1 + n_2 - 2} }$

Cohen's $d_s$ is then:

$d_s = \frac{m_1 - m_2}{s_{pooled}}$

The corresponding t-statistic is:

$t = \frac{m_1 - m_2}{ \sqrt{ s_{pooled}^2/n_1 + s_{pooled}^2/n_2 }}$

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d_s$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d_s$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d_s$.

M1, M2

Group means.

sd1, sd2

Standard deviations for each group.

se1, se2

Standard errors for each group mean.

M1low, M1high, M2low, M2high

Confidence interval bounds for each group mean.

spooled

Pooled standard deviation.

sepooled

Pooled standard error.

n1, n2

Group sample sizes.

df

Degrees of freedom ($n_1 - 1 + n_2 - 1$).

t

t-statistic.

p

p-value.

estimate

APA-style formatted string for reporting $d_s$ and its CI.

statistic

APA-style formatted string for reporting the t-statistic and p-value.

Examples

# The following example is derived from the "indt_data" dataset
# included in MOTE.

# A forensic psychologist examined whether being hypnotized during recall
# affects how well a witness remembers facts about an event.

t.test(correctq ~ group, data = indt_data)

# Direct entry of summary statistics:
d_ind_t(m1 = 17.75, m2 = 23, sd1 = 3.30,
        sd2 = 2.16, n1 = 4, n2 = 4, a = .05)

# Equivalent shorthand:
d_ind_t(17.75, 23, 3.30, 2.16, 4, 4, .05)

# Using raw data from the dataset:
d_ind_t(mean(indt_data$correctq[indt_data$group == 1]),
        mean(indt_data$correctq[indt_data$group == 2]),
        sd(indt_data$correctq[indt_data$group == 1]),
        sd(indt_data$correctq[indt_data$group == 2]),
        length(indt_data$correctq[indt_data$group == 1]),
        length(indt_data$correctq[indt_data$group == 2]),
        .05)

---

Cohen's d from t for Independent Samples (Pooled SD)

Description

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'd_ind_t_t()' to follow modern R style guidelines. The original dotted version 'd.ind.t.t()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'd', 'dlow', 'dhigh', 'n1', 'n2', 'df', 't', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'd_lower_limit', 'd_upper_limit', 'sample_size_1', 'sample_size_2', 'degrees_freedom', 't', 'p_value'). New code should prefer 'd_ind_t_t()' and the snake_case output names, but existing code using the older names will continue to work.

Usage

d_ind_t_t(t_value, t = NULL, n1, n2, a = 0.05)

d.ind.t.t(t, n1, n2, a = 0.05)

Arguments

t_value	t-statistic from an independent-samples t-test.
t	t-statistic from an independent-samples t-test. Used for backwards compatibility.
n1	Sample size for group one.
n2	Sample size for group two.
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

Compute Cohen's $d_s$ from an independent-samples t-statistic and provide a noncentral-t confidence interval, assuming equal variances (pooled SD).

For between-subjects designs with pooled SD, $d_s$ can be obtained directly from the t-statistic:

$d_s = \frac{2t}{\sqrt{n_1 + n_2 - 2}},$

where $n_1$ and $n_2$ are the group sample sizes (df = $n_1 + n_2 - 2$). The $(1-\alpha)$ confidence interval for $d_s$ is derived from the noncentral t distribution for the observed $t$ and df.

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d_s$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d_s$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d_s$.

n1, n2

Group sample sizes.

df

Degrees of freedom ($n_1 + n_2 - 2$).

t

t-statistic.

p

p-value.

estimate

APA-style formatted string for reporting $d_s$ and its CI.

statistic

APA-style formatted string for reporting the t-statistic and p-value.

Examples

# The following example is derived from the "indt_data" dataset in MOTE.
    hyp <- t.test(correctq ~ group, data = indt_data)

# Direct entry of the t-statistic and sample sizes:
    d_ind_t_t(t = -2.6599, n1 = 4, n2 = 4, a = .05)

# Using the t-statistic from the model object:
    d_ind_t_t(hyp$statistic, length(indt_data$group[indt_data$group == 1]),
              length(indt_data$group[indt_data$group == 2]), .05)

---

Cohen's d (SMD) for Independent Proportions (Binary Outcomes)

Description

This function computes a standardized mean difference effect size for two independent proportions by treating each as the mean of a Bernoulli (0/1) variable and computing a standardized mean difference (SMD) directly using the pooled Bernoulli standard deviation. This follows the same logic as Cohen's d for continuous variables, but applied to binary outcomes:

Usage

d_prop(p1, p2, n1, n2, a = 0.05)

d.prop(p1, p2, n1, n2, a = 0.05)

Arguments

p1	Proportion for group one (between 0 and 1).
p2	Proportion for group two (between 0 and 1).
n1	Sample size for group one.
n2	Sample size for group two.
a	Significance level used for confidence intervals. Defaults to 0.05.

Details

$d = \frac{p_1 - p_2}{s_{\mathrm{pooled}}}$

where

$s_{\mathrm{pooled}} = \sqrt{\frac{(n_1 - 1)p_1(1 - p_1) + (n_2 - 1)p_2(1 - p_2)} {n_1 + n_2 - 2}}$

This replaces the original z‐based formulation used in older versions of MOTE. The SMD effect size is directly comparable to all other d‐type effect sizes in the package.

Value

A list with the same structure as [d_ind_t()], containing the standardized mean difference and its confidence interval, along with auxiliary statistics. The list is augmented with explicit entries 'p1', 'p2', 'p1_value', and 'p2_value' to emphasize that the original inputs were proportions.

Examples

d_prop(p1 = .25, p2 = .35, n1 = 100, n2 = 100, a = .05)

---

Cohen's d for One-Sample t from Summary Stats

Description

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'd_single_t()' to follow modern R style guidelines. The original dotted version 'd.single.t()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'd', 'dlow', 'dhigh', 'm', 'sd', 'se', 'Mlow', 'Mhigh', 'u', 'n', 'df', 't', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'd_lower_limit', 'd_upper_limit', 'mean_value', 'sd_value', 'se_value', 'mean_lower_limit', 'mean_upper_limit', 'population_mean', 'sample_size', 'degrees_freedom', 't_value', 'p_value'). New code should prefer 'd_single_t()' and the snake_case output names, but existing code using the older names will continue to work.

Usage

d_single_t(m, u, sd, n, a = 0.05)

d.single.t(m, u, sd, n, a = 0.05)

Arguments

m	Sample mean.
u	Population (reference) mean $\mu$.
sd	Sample standard deviation $s$.
n	Sample size $n$.
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

Compute Cohen's $d$ and a noncentral-t confidence interval for a one-sample (single) t-test using summary statistics.

The effect size is defined as the standardized mean difference between the sample mean and the population/reference mean:

$d = \frac{m - \mu}{s}.$

The corresponding t-statistic is:

$t = \frac{m - \mu}{s/\sqrt{n}}.$

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d$.

m

Sample mean.

sd

Sample standard deviation.

se

Standard error of the mean.

Mlow, Mhigh

Confidence interval bounds for the mean.

u

Population (reference) mean.

n

Sample size.

df

Degrees of freedom ($n - 1$).

t

t-statistic.

p

p-value.

estimate

APA-style formatted string for reporting $d$ and its CI.

statistic

APA-style formatted string for reporting the t-statistic and p-value.

Examples

# Example derived from the "singt_data" dataset included in MOTE.

# A school claims their gifted/honors program outperforms the national
# average (1080). Their students' SAT scores (sample) have mean 1370 and
# SD 112.7.

    gift <- t.test(singt_data$SATscore, mu = 1080, alternative = "two.sided")

# Direct entry of summary statistics:
    d_single_t(m = 1370, u = 1080, sd = 112.7, n = 14, a = .05)

# Equivalent shorthand:
    d_single_t(1370, 1080, 112.7, 14, .05)

# Using values from the t-test object and dataset:
    d_single_t(gift$estimate, gift$null.value,
               sd(singt_data$SATscore), length(singt_data$SATscore), .05)

---

Cohen's d from t for One-Sample t-Test

Description

Compute Cohen's $d$ and a noncentral-t confidence interval for a one-sample (single) t-test using the observed t-statistic.

Usage

d_single_t_t(t, n, a = 0.05)

d.single.t.t(t, n, a = 0.05)

Arguments

t	t-test value.
n	Sample size.
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).

Details

The effect size is calculated as:

$d = \frac{t}{\sqrt{n}},$

where $t$ is the one-sample t-statistic and $n$ is the sample size.

The corresponding $(1 - \alpha)$ confidence interval for $d$ is derived from the noncentral t distribution.

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Cohen's $d$.

dlow

Lower limit of the $(1-\alpha)$ confidence interval for $d$.

dhigh

Upper limit of the $(1-\alpha)$ confidence interval for $d$.

n

Sample size.

df

Degrees of freedom ($n - 1$).

t

t-statistic.

p

p-value.

estimate

APA-style formatted string for reporting $d$ and its CI.

statistic

APA-style formatted string for reporting the t-statistic and p-value.

Examples

# A school has a gifted/honors program that they claim is
# significantly better than others in the country. The gifted/honors
# students in this school scored an average of 1370 on the SAT,
# with a standard deviation of 112.7, while the national average
# for gifted programs is a SAT score of 1080.

    gift <- t.test(singt_data$SATscore, mu = 1080, alternative = "two.sided")

# Direct entry of t-statistic and sample size:
    d_single_t_t(9.968, 15, .05)

# Equivalent shorthand:
    d_single_t_t(9.968, 15, .05)

# Using values from a t-test object and dataset:
    d_single_t_t(gift$statistic, length(singt_data$SATscore), .05)

---

r and Coefficient of Determination (R2) from d

Description

**Note on function and output names:** This effect size translation is now implemented with the snake_case function name 'd_to_r()' to follow modern R style guidelines. The original dotted version 'd.to.r()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'r', 'rlow', 'rhigh', 'R2', 'R2low', 'R2high', 'se', 'n', 'dfm', 'dfe', 't', 'F', 'p', 'estimate', 'estimateR2', 'statistic') and newer snake_case aliases (e.g., 'r_lower_limit', 'r_upper_limit', 'r2_value', 'r2_lower_limit', 'r2_upper_limit', 'se_value', 'sample_size', 'degrees_freedom_model', 'degrees_freedom_error', 't_value', 'f_value', 'p_value', 'estimate_r', 'estimate_r2'). New code should prefer 'd_to_r()' and the snake_case output names, but existing code using the older names will continue to work.

Usage

d_to_r(d, n1, n2, a = 0.05)

d.to.r(d, n1, n2, a = 0.05)

Arguments

d	Effect size statistic.
n1	Sample size for group one.
n2	Sample size for group two.
a	Significance level.

Details

Calculates r from d and then translates r to r2 to calculate the non-central confidence interval for r2 using the F distribution.

The correlation coefficient ($r$) is calculated by dividing Cohen's d by the square root of the total sample size squared, divided by the product of the sample sizes of group one and group two.

$r = \frac{d}{\sqrt{d^2 + \frac{(n_1 + n_2)^2}{n_1 n_2}}}$

Learn more on our example page.

Value

Provides the effect size (correlation coefficient) with associated confidence intervals, the t-statistic, F-statistic, and other estimates appropriate for d to r translation. Note this CI is not based on the traditional r-to-z transformation but rather non-central F using the ci.R function from MBESS.

r

Correlation coefficient.

rlow

Lower level confidence interval for r.

rhigh

Upper level confidence interval for r.

R2

Coefficient of determination.

R2low

Lower level confidence interval of R2.

R2high

Upper level confidence interval of R2.

se

Standard error.

n

Sample size.

dfm

Degrees of freedom of mean.

dfe

Degrees of freedom error.

t

t-statistic.

F

F-statistic.

p

p-value.

estimate

The r statistic and confidence interval in APA style for markdown printing.

estimateR2

The R$^2$ statistic and confidence interval in APA style for markdown printing.

statistic

The t-statistic in APA style for markdown printing.

Examples

# The following example is derived from the "indt_data"
# dataset, included in the MOTE library.

# A forensic psychologist conducted a study to examine whether
# being hypnotized during recall affects how well a witness
# can remember facts about an event. Eight participants
# watched a short film of a mock robbery, after which
# each participant was questioned about what he or she had
# seen. The four participants in the experimental group
# were questioned while they were hypnotized. The four
# participants in the control group received the same
# questioning without hypnosis.

# Contrary to the hypothesized result, the group that underwent
# hypnosis were significantly less accurate while reporting
# facts than the control group with a large effect size, t(6) = -2.66,
# p = .038, d_s = -1.88.

     d_to_r(d = -1.88, n1 = 4, n2 = 4, a = .05)

---

Cohen's d for Z-test from Population Mean and SD

Description

Computes Cohen's d for a Z-test using the sample mean, population mean, and population standard deviation. The function also provides a normal-theory confidence interval for d, and returns relevant statistics including the z-statistic and its p-value.

Usage

d_z_mean(mu, m1, sig, sd1, n, a = 0.05)

d.z.mean(mu, m1, sig, sd1, n, a = 0.05)

Arguments

mu	The population mean.
m1	The sample study mean.
sig	The population standard deviation.
sd1	The standard deviation from the study.
n	The sample size.
a	The significance level.

Details

The effect size is computed as:

$d = \frac{m_1 - \mu}{\sigma}$

where $m_1$ is the sample mean, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

The z-statistic is:

$z = \frac{m_1 - \mu}{\sigma / \sqrt{n}}$

where $n$ is the sample size.

Learn more on our example page.

Value

A list with the following components:

d

Effect size (Cohen's d).

dlow

Lower level confidence interval d value.

dhigh

Upper level confidence interval d value.

M1

Mean of sample.

sd1

Standard deviation of sample.

se1

Standard error of sample.

M1low

Lower level confidence interval of the mean.

M1high

Upper level confidence interval of the mean.

Mu

Population mean.

Sigma

Standard deviation of population.

se2

Standard error of population.

z

Z-statistic.

p

P-value.

n

Sample size.

estimate

The d statistic and confidence interval in APA style for markdown printing.

statistic

The Z-statistic in APA style for markdown printing.

Examples

# The average quiz test taking time for a 10 item test is 22.5
# minutes, with a standard deviation of 10 minutes. My class of
# 25 students took 19 minutes on the test with a standard deviation of 5.

d_z_mean(mu = 22.5, m1 = 19, sig = 10, sd1 = 5, n = 25, a = .05)

---

Cohen's d from z-statistic for Z-test

Description

Compute Cohen's $d$ from a z-statistic for a Z-test.

Usage

d_z_z(z, n, a = 0.05, sig = NA)

d.z.z(z, sig = NA, n, a = 0.05)

Arguments

z	z-statistic from a Z-test.
n	Sample size.
a	Significance level (alpha) for the confidence interval. Must be in (0, 1).
sig	Population standard deviation ($\sigma$). This value is retained for descriptive purposes but is not required to compute the confidence interval for $d$.

Details

The effect size is computed as:

$d = \frac{z}{\sqrt{n}},$

where $n$ is the sample size.

The confidence interval bounds assume a normal-theory standard error for $d$ of $1 / \sqrt{n}$ (given that $d = z / \sqrt{n}$). Thus:

$d_{\mathrm{low}} = d - z_{\alpha/2} \cdot 1/\sqrt{n}$

$d_{\mathrm{high}} = d + z_{\alpha/2} \cdot 1/\sqrt{n}$

where $z_{\alpha/2}$ is the critical value from the standard normal distribution.

The population standard deviation ($\sigma$) is retained for descriptive purposes but is not required for computing confidence intervals for $d$.

See the online example for additional context: Learn more on our example page.

Value

A list with the following elements:

d

Effect size.

dlow

Lower confidence interval bound for $d$.

dhigh

Upper confidence interval bound for $d$.

sigma

Population standard deviation ($\sigma$).

z

z-statistic.

p

Two-tailed p-value.

n

Sample size.

estimate

The $d$ statistic and confidence interval in APA style for markdown printing.

statistic

The Z-statistic in APA style for markdown printing.

Examples

# A recent study suggested that students (N = 100) learning
# statistics improved their test scores with the use of
# visual aids (Z = 2.5). The population standard deviation is 4.

# You can type in the numbers directly as shown below,
# or refer to your dataset within the function.

    d_z_z(z = 2.5, sig = 4, n = 100, a = .05)

    d_z_z(z = 2.5, n = 100, a = .05)

    d.z.z(2.5, 4, 100, .05)

---

$d_{\delta}$ for Between Subjects with Control Group SD Denominator

Description

This function displays $d_{\delta}$ for between subjects data and the non-central confidence interval using the control group standard deviation as the denominator.

Usage

delta_ind_t(m1, m2, sd1, sd2, n1, n2, a = 0.05)

delta.ind.t(m1, m2, sd1, sd2, n1, n2, a = 0.05)

Arguments

m1	mean from control group
m2	mean from experimental group
sd1	standard deviation from control group
sd2	standard deviation from experimental group
n1	sample size from control group
n2	sample size from experimental group
a	significance level

Details

To calculate $d_{\delta}$, the mean of the experimental group is subtracted from the mean of the control group, which is divided by the standard deviation of the control group.

$d_{\delta} = \frac{m_1 - m_2}{sd_1}$

Learn more on our example page.

Value

Provides the effect size (Cohen's d) with associated confidence intervals, the t-statistic, the confidence intervals associated with the means of each group, as well as the standard deviations and standard errors of the means for each group.

d	d-delta effect size
dlow	lower level confidence interval of d-delta value
dhigh	upper level confidence interval of d-delta value
M1	mean of group one
sd1	standard deviation of group one mean
se1	standard error of group one mean
M1low	lower level confidence interval of group one mean
M1high	upper level confidence interval of group one mean
M2	mean of group two
sd2	standard deviation of group two mean
se2	standard error of group two mean
M2low	lower level confidence interval of group two mean
M2high	upper level confidence interval of group two mean
spooled	pooled standard deviation
sepooled	pooled standard error
n1	sample size of group one
n2	sample size of group two
df	degrees of freedom (n1 - 1 + n2 - 1)
t	t-statistic
p	p-value
estimate	the d statistic and confidence interval in APA style for markdown printing
statistic	the t-statistic in APA style for markdown printing

Examples

# The following example is derived from the "indt_data"
# dataset, included in the MOTE library.

# A forensic psychologist conducted a study to examine whether
# being hypnotized during recall affects how well a witness
# can remember facts about an event. Eight participants
# watched a short film of a mock robbery, after which
# each participant was questioned about what he or she had
# seen. The four participants in the experimental group
# were questioned while they were hypnotized. The four
# participants in the control group received the same
# questioning without hypnosis.

    hyp <- t.test(correctq ~ group, data = indt_data)

# You can type in the numbers directly, or refer to the dataset,
# as shown below.

    delta_ind_t(m1 = 17.75, m2 = 23,
               sd1 = 3.30, sd2 = 2.16,
                n1 = 4, n2 = 4, a = .05)

    delta_ind_t(17.75, 23, 3.30, 2.16, 4, 4, .05)

    delta_ind_t(mean(indt_data$correctq[indt_data$group == 1]),
            mean(indt_data$correctq[indt_data$group == 2]),
            sd(indt_data$correctq[indt_data$group == 1]),
            sd(indt_data$correctq[indt_data$group == 2]),
            length(indt_data$correctq[indt_data$group == 1]),
            length(indt_data$correctq[indt_data$group == 2]),
            .05)

# Contrary to the hypothesized result, the group that underwent hypnosis were
# significantly less accurate while reporting facts than the control group
# with a large effect size, t(6) = -2.66, p = .038, d_delta = 1.59.

---

Dependent t Example Data

Description

Dataset for use in d_dep_t_diff, d_dep_t_diff_t, d_dep_t_avg, and d_dep_t_rm exploring the before and after effects of scifi movies on supernatural beliefs.

Usage

data(dept_data)

Format

A data frame of before and after scores for rating supernatural beliefs.

before: scores rated before watching a scifi movie after: scores rated after watching a scifi movie

References

Nolan and Heizen Statistics for the Behavioral Sciences

---

$\epsilon^2$ for ANOVA from $F$ and Sum of Squares

Description

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'epsilon_full_ss()' to follow modern R style guidelines. The original dotted version 'epsilon.full.SS()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'epsilon', 'epsilonlow', 'epsilonhigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'epsilon_value', 'epsilon_lower_limit', 'epsilon_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'epsilon_full_ss()' and the snake_case output names, but existing code using the older names will continue to work.

Usage

epsilon_full_ss(dfm, dfe, msm, mse, sst, a = 0.05)

epsilon.full.SS(dfm, dfe, msm, mse, sst, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
msm	mean square for the model/IV/between
mse	mean square for the error/residual/within
sst	sum of squares total
a	significance level

Details

This function displays $\epsilon^2$ from ANOVA analyses and its non-central confidence interval based on the $F$ distribution. This formula works for one way and multi way designs with careful focus on the sum of squares total calculation.

To calculate $\epsilon^2$, first, the mean square for the error is is multiplied by the degrees of freedom for the model. The product is divided by the sum of squares total.

$\epsilon^2 = \frac{df_m (ms_m - ms_e)}{SS_T}$

Learn more on our example page.

Value

Provides the effect size ($\epsilon^2$) with associated confidence intervals from the $F$-statistic.

epsilon

effect size

epsilonlow

lower level confidence interval of epsilon

epsilonhigh

upper level confidence interval of epsilon

dfm

degrees of freedom for the model/IV/between

dfe

degrees of freedom for the error/residual/within

F

$F$-statistic

p

p-value

estimate

the $\epsilon^2$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

Examples

# The following example is derived from the "bn1_data"
# dataset, included in the MOTE library.

# A health psychologist recorded the number of close inter-personal
# attachments of 45-year-olds who were in excellent, fair, or poor
# health. People in the Excellent Health group had 4, 3, 2, and 3
# close attachments; people in the Fair Health group had 3, 5,
# and 8 close attachments; and people in the Poor Health group
# had 3, 1, 0, and 2 close attachments.

anova_model <- lm(formula = friends ~ group, data = bn1_data)
summary.aov(anova_model)

epsilon_full_ss(dfm = 2, dfe = 8, msm = 12.621,
                mse = 2.458, sst = (25.24 + 19.67), a = .05)

# Backwards-compatible dotted name (deprecated)
epsilon.full.SS(dfm = 2, dfe = 8, msm = 12.621,
                mse = 2.458, sst = (25.24 + 19.67), a = .05)

---

$\eta^2$ and Coefficient of Determination (R$^2$) for ANOVA from $F$

Description

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'eta_f()' to follow modern R style guidelines. The original dotted version 'eta.F()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'eta', 'etalow', 'etahigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'eta_value', 'eta_lower_limit', 'eta_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'eta_f()' and the snake_case output names, but existing code using the older names will continue to work.

Usage

eta_f(dfm, dfe, f_value, a = 0.05, Fvalue)

eta.F(dfm, dfe, Fvalue, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
f_value	F statistic
a	significance level
Fvalue	F statistic only for older function

Details

This function displays $\eta^2$ from ANOVA analyses and their non-central confidence interval based on the $F$ distribution. These values are calculated directly from F statistics and can be used for between subjects and repeated measures designs. Remember if you have two or more IVs, these values are partial eta squared.

Eta is calculated by multiplying the degrees of freedom of the model by the F-statistic. This is divided by the product of degrees of freedom of the model, the F-statistic, and the degrees of freedom for the error or residual.

$\eta^2 = \frac{df_m \cdot F}{df_m \cdot F + df_e}$

Learn more on our example page.

Value

Provides the effect size ($\eta^2$) with associated confidence intervals and relevant statistics.

eta

$\eta^2$ effect size

etalow

lower level confidence interval of $\eta^2$

etahigh

upper level confidence interval of $\eta^2$

dfm

degrees of freedom for the model/IV/between

dfe

degrees of freedom for the error/residual/within

F

$F$-statistic

p

p-value

estimate

the $\eta^2$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

Examples

# The following example is derived from the "bn1_data"
# dataset, included in the MOTE library.

# A health psychologist recorded the number of close inter-personal
# attachments of 45-year-olds who were in excellent, fair, or poor
# health. People in the Excellent Health group had 4, 3, 2, and 3
# close attachments; people in the Fair Health group had 3, 5,
# and 8 close attachments; and people in the Poor Health group
# had 3, 1, 0, and 2 close attachments.

anova_model <- lm(formula = friends ~ group, data = bn1_data)
summary.aov(anova_model)

eta_f(dfm = 2, dfe = 8,
      Fvalue = 5.134, a = .05)

# Backwards-compatible dotted name (deprecated)
eta.F(dfm = 2, dfe = 8,
      Fvalue = 5.134, a = .05)

---

$\eta^2$ for ANOVA from $F$ and Sum of Squares

Description

This function displays $\eta^2$ from ANOVA analyses and its non-central confidence interval based on the $F$ distribution. This formula works for one way and multi way designs with careful focus on the sum of squares total.

Usage

eta_full_ss(dfm, dfe, ssm, sst, f_value, a = 0.05, Fvalue)

eta.full.SS(dfm, dfe, ssm, sst, Fvalue, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
ssm	sum of squares for the model/IV/between
sst	sum of squares total
f_value	F statistic
a	significance level
Fvalue	Backward-compatible argument for the F statistic (deprecated; use 'f_value' instead). If supplied, it overrides 'f_value'. Included for users of the legacy 'eta.full.SS()'.

Details

Eta squared is calculated by dividing the sum of squares for the model by the sum of squares total.

$\eta^2 = \frac{SS_M}{SS_T}$

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'eta_full_ss()' to follow modern R style guidelines. The original dotted version 'eta.full.SS()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'eta', 'etalow', 'etahigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'eta_value', 'eta_lower_limit', 'eta_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'eta_full_ss()' and the snake_case output names, but existing code using the older names will continue to work.

Learn more on our example page.

Value

Provides the effect size ($\eta^2$) with associated confidence intervals and relevant statistics.

eta

$\eta^2$ effect size

etalow

lower level confidence interval of $\eta^2$

etahigh

upper level confidence interval of $\eta^2$

dfm

degrees of freedom for the model/IV/between

dfe

degrees of freedom for the error/residual/within

F

$F$-statistic

p

p-value

estimate

the $\eta^2$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

Examples

# The following example is derived from the "bn1_data"
# dataset, included in the MOTE library.

# A health psychologist recorded the number of close inter-personal
# attachments of 45-year-olds who were in excellent, fair, or poor
# health. People in the Excellent Health group had 4, 3, 2, and 3
# close attachments; people in the Fair Health group had 3, 5,
# and 8 close attachments; and people in the Poor Health group
# had 3, 1, 0, and 2 close attachments.

anova_model <- lm(formula = friends ~ group, data = bn1_data)
summary.aov(anova_model)

eta_full_ss(dfm = 2, dfe = 8, ssm = 25.24,
            sst = (25.24 + 19.67), f_value = 5.134, a = .05)

# Backwards-compatible dotted name (deprecated)
eta.full.SS(dfm = 2, dfe = 8, ssm = 25.24,
            sst = (25.24 + 19.67), Fvalue = 5.134, a = .05)

---

$\eta^2_p$ for ANOVA from $F$ and Sum of Squares

Description

This function displays $\eta^2_p$ from ANOVA analyses and its non-central confidence interval based on the $F$ distribution. This formula works for one way and multi way designs.

Usage

eta_partial_ss(dfm, dfe, ssm, sse, f_value, a = 0.05, Fvalue)

eta.partial.SS(dfm, dfe, ssm, sse, Fvalue, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
ssm	sum of squares for the model/IV/between
sse	sum of squares for the error/residual/within
f_value	F statistic
a	significance level
Fvalue	Backward-compatible argument for the F statistic (deprecated; use 'f_value' instead). If supplied, it overrides 'f_value'. Included for users of the legacy 'eta.partial.SS()'.

Details

$\eta^2_p$ is calculated by dividing the sum of squares of the model by the sum of the sum of squares of the model and sum of squares of the error.

$\eta^2_p = \frac{SS_M}{SS_M + SS_E}$

Learn more on our example page.

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'eta_partial_ss()' to follow modern R style guidelines. The original dotted version 'eta.partial.SS()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'eta', 'etalow', 'etahigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'eta_value', 'eta_lower_limit', 'eta_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'eta_partial_ss()' and the snake_case output names, but existing code using the older names will continue to work.

Value

Provides the effect size ($\eta^2_p$) with associated confidence intervals and relevant statistics.

eta

$\eta^2_p$ effect size

etalow

lower level confidence interval of $\eta^2_p$

etahigh

upper level confidence interval of $\eta^2_p$

dfm

degrees of freedom for the model/IV/between

dfe

degrees of freedom for the error/residual/within

F

$F$-statistic

p

p-value

estimate

the $\eta^2_p$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

Examples

# The following example is derived from the "bn2_data"
# dataset, included in the MOTE library.

# Is there a difference in athletic spending budget for different sports?
# Does that spending interact with the change in coaching staff?
# This data includes (fake) athletic budgets for baseball, basketball,
# football, soccer, and volleyball teams with new and old coaches
# to determine if there are differences in
# spending across coaches and sports.

# Example using reported ANOVA table values directly
eta_partial_ss(dfm = 4, dfe = 990,
               ssm = 338057.9, sse = 32833499,
               f_value = 2.548, a = .05)

# Example computing Type III SS with code (requires the "car" package)
if (requireNamespace("car", quietly = TRUE)) {

  # Fit the model using stats::lm
  mod <- stats::lm(money ~ coach * type, data = bn2_data)

  # Type III table for the effects
  aov_type3 <- car::Anova(mod, type = 3)

  # Extract DF, SS, and F for the interaction (coach:type)
  dfm_int <- aov_type3["coach:type", "Df"]
  ssm_int <- aov_type3["coach:type", "Sum Sq"]
  F_int   <- aov_type3["coach:type", "F value"]

  # Residual DF and SS from the standard ANOVA table
  aov_type1 <- stats::anova(mod)
  dfe <- aov_type1["Residuals", "Df"]
  sse <- aov_type1["Residuals", "Sum Sq"]

  # Calculate partial eta-squared for the interaction using Type III SS
  eta_partial_ss(dfm = dfm_int, dfe = dfe,
                 ssm = ssm_int, sse = sse,
                 f_value = F_int, a = .05)
#'
# Backwards-compatible dotted name (deprecated)
eta.partial.SS(dfm = 4, dfe = 990,
               ssm = 338057.9, sse = 32833499,
               Fvalue = 2.548, a = .05)
}

---

$d_g$ Corrected for Independent $t$

Description

This function displays $d_g$ (Hedges' g) corrected and the non-central confidence interval for independent $t$.

Usage

g_ind_t(m1, m2, sd1, sd2, n1, n2, a = 0.05)

g.ind.t(m1, m2, sd1, sd2, n1, n2, a = 0.05)

Arguments

m1	mean group one
m2	mean group two
sd1	standard deviation group one
sd2	standard deviation group two
n1	sample size group one
n2	sample size group two
a	significance level

Details

The small-sample correction factor is:

$\mathrm{correction} = 1 - \frac{3}{4(n_1 + n_2) - 9}$

$d_g$ is computed as the standardized mean difference multiplied by the correction:

$d_g = \frac{m_1 - m_2}{s_{\mathrm{pooled}}} \times \mathrm{correction}$

Learn more on our example page.

Value

d

$d_g$ corrected effect size

dlow

lower level confidence interval for $d_g$

dhigh

upper level confidence interval for $d_g$

M1

mean of group one

sd1

standard deviation of group one

se1

standard error of group one

M1low

lower level confidence interval of mean one

M1high

upper level confidence interval of mean one

M2

mean of group two

sd2

standard deviation of group two

se2

standard error of group two

M2low

lower level confidence interval of mean two

M2high

upper level confidence interval of mean two

spooled

pooled standard deviation

sepooled

pooled standard error

correction

Hedges' small-sample correction factor

n1

sample size of group one

n2

sample size of group two

df

degrees of freedom ($n_1 - 1 + n_2 - 1$)

t

$t$-statistic

p

p-value

estimate

the $d_g$ statistic and confidence interval in APA style for markdown printing

statistic

the $t$-statistic in APA style for markdown printing

Examples

# The following example is derived from the "indt_data"
# dataset, included in the MOTE library.

# A forensic psychologist conducted a study to examine whether
# being hypnotized during recall affects how well a witness
# can remember facts about an event. Eight participants
# watched a short film of a mock robbery, after which
# each participant was questioned about what he or she had
# seen. The four participants in the experimental group
# were questioned while they were hypnotized. The four
# participants in the control group received the same
# questioning without hypnosis.

   t.test(correctq ~ group, data = indt_data)

# You can type in the numbers directly, or refer to the dataset,
# as shown below.

    g_ind_t(m1 = 17.75, m2 = 23, sd1 = 3.30,
           sd2 = 2.16, n1 = 4, n2 = 4, a = .05)

    g_ind_t(17.75, 23, 3.30, 2.16, 4, 4, .05)

    g_ind_t(mean(indt_data$correctq[indt_data$group == 1]),
            mean(indt_data$correctq[indt_data$group == 2]),
            sd(indt_data$correctq[indt_data$group == 1]),
            sd(indt_data$correctq[indt_data$group == 2]),
            length(indt_data$correctq[indt_data$group == 1]),
            length(indt_data$correctq[indt_data$group == 2]),
            .05)

# Contrary to the hypothesized result, the group that underwent hypnosis were
# significantly less accurate while reporting facts than the control group
# with a large effect size, t(6) = -2.66, p = .038, d_g = 1.64.

---

$\eta^2_{G}$ (Partial Generalized Eta-Squared) for Mixed Design ANOVA from $F$

Description

This function displays partial generalized eta-squared ($\eta^2_{G}$) from ANOVA analyses and its non-central confidence interval based on the $F$ distribution. This formula works for mixed designs.

Usage

ges_partial_ss_mix(dfm, dfe, ssm, sss, sse, f_value, a = 0.05, Fvalue)

ges.partial.SS.mix(dfm, dfe, ssm, sss, sse, Fvalue, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
ssm	sum of squares for the model/IV/between
sss	sum of squares subject variance
sse	sum of squares for the error/residual/within
f_value	F statistic
a	significance level
Fvalue	Backward-compatible argument for the F statistic (deprecated; use 'f_value' instead). If supplied, it overrides 'f_value'. Included for users of the legacy 'ges.partial.SS.mix()' API.

Details

To calculate partial generalized eta squared, first, the sum of squares of the model, sum of squares of the subject variance, sum of squares for the subject variance, and the sum of squares for the error/residual/within are added together.

$\eta^2_{G} = \frac{SS_M}{SS_M + SS_S + SS_E}$

Learn more on our example page.

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'ges_partial_ss_mix()' to follow modern R style guidelines. The original dotted version 'ges.partial.SS.mix()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'ges', 'geslow', 'geshigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'ges_value', 'ges_lower_limit', 'ges_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'ges_partial_ss_mix()' and the snake_case output names, but existing code using the older names will continue to work.

Value

ges

$\eta^2_{G}$ effect size

geslow

lower level confidence interval for $\eta^2_{G}$

geshigh

upper level confidence interval for $\eta^2_{G}$

dfm

degrees of freedom for the model/IV/between

dfe

degrees of freedom for the error/residual/within

F

$F$-statistic

p

p-value

estimate

the $\eta^2_{G}$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

Examples

# The following example is derived from the
# "mix2_data" dataset, included in the MOTE library.

# Given previous research, we know that backward strength in free
# association tends to increase the ratings participants give when
# you ask them how many people out of 100 would say a word in
# response to a target word (like Family Feud). This result is
# tied to people’s overestimation of how well they think they know
# something, which is bad for studying. So, we gave people instructions
# on how to ignore the BSG.  Did it help? Is there an interaction
# between BSG and instructions given?

# You would calculate one partial GES value for each F-statistic.
# Here's an example for the interaction using reported ANOVA values.
ges_partial_ss_mix(dfm = 1, dfe = 156,
                   ssm = 71.07608,
                   sss = 30936.498,
                   sse = 8657.094,
                   f_value = 1.280784, a = .05)

# Backwards-compatible dotted name (deprecated)
ges.partial.SS.mix(dfm = 1, dfe = 156,
                   ssm = 71.07608,
                   sss = 30936.498,
                   sse = 8657.094,
                   Fvalue = 1.280784, a = .05)

---

$\eta^2_{G}$ (Partial Generalized Eta-Squared) for Repeated-Measures ANOVA from $F$

Description

This function displays partial generalized eta-squared ($\eta^2_{G}$) from ANOVA analyses and its non-central confidence interval based on the $F$ distribution. This formula works for multi-way repeated measures designs.

Usage

ges_partial_ss_rm(
  dfm,
  dfe,
  ssm,
  sss,
  sse1,
  sse2,
  sse3,
  f_value,
  a = 0.05,
  Fvalue
)

ges.partial.SS.rm(dfm, dfe, ssm, sss, sse1, sse2, sse3, Fvalue, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
ssm	sum of squares for the model/IV/between
sss	sum of squares subject variance
sse1	sum of squares for the error/residual/within for the first IV
sse2	sum of squares for the error/residual/within for the second IV
sse3	sum of squares for the error/residual/within for the interaction
f_value	F statistic
a	significance level
Fvalue	Backward-compatible argument for the F statistic (deprecated; use 'f_value' instead). If supplied, it overrides 'f_value'. Included for users of the legacy 'ges.partial.SS.rm()' API.

Details

To calculate partial generalized eta squared, first, the sum of squares of the model, sum of squares of the subject variance, sum of squares for the first and second independent variables, and the sum of squares for the interaction are added together. The sum of squares of the model is divided by this value.

$\eta^2_{G} = \frac{SS_M}{SS_M + SS_S + SS_{E1} + SS_{E2} + SS_{E3}}$

Learn more on our example page.

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'ges_partial_ss_rm()' to follow modern R style guidelines. The original dotted version 'ges.partial.SS.rm()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'ges', 'geslow', 'geshigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'ges_value', 'ges_lower_limit', 'ges_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'ges_partial_ss_rm()' and the snake_case output names, but existing code using the older names will continue to work.

Value

ges

$\eta^2_{G}$ effect size

geslow

lower level confidence interval for $\eta^2_{G}$

geshigh

upper level confidence interval for $\eta^2_{G}$

dfm

degrees of freedom for the model/IV/between

dfe

degrees of freedom for the error/residual/within

F

$F$-statistic

p

p-value

estimate

the $\eta^2_{G}$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

Examples

# The following example is derived from the "rm2_data" dataset, included
# in the MOTE library.

# In this experiment people were given word pairs to rate based on
# their "relatedness". How many people out of a 100 would put LOST-FOUND
# together? Participants were given pairs of words and asked to rate them
# on how often they thought 100 people would give the second word if shown
# the first word.  The strength of the word pairs was manipulated through
# the actual rating (forward strength: FSG) and the strength of the reverse
# rating (backward strength: BSG). Is there an interaction between FSG and
# BSG when participants are estimating the relation between word pairs?

# You would calculate one partial GES value for each F-statistic.
# Here's an example for the interaction with typing in numbers.
ges_partial_ss_rm(dfm = 1, dfe = 157,
                  ssm = 2442.948, sss = 76988.13,
                  sse1 = 5402.567, sse2 = 8318.75, sse3 = 6074.417,
                  f_value = 70.9927, a = .05)

# Backwards-compatible dotted name (deprecated)
ges.partial.SS.rm(dfm = 1, dfe = 157,
                  ssm = 2442.948, sss = 76988.13,
                  sse1 = 5402.567, sse2 = 8318.75, sse3 = 6074.417,
                  Fvalue = 70.9927, a = .05)

---

Cohen's h for Independent Proportions

Description

This function computes Cohen's $h$ effect size for the difference between two independent proportions. Cohen's $h$ is defined as a difference between arcsine-transformed proportions:

Usage

h_prop(p1, p2, n1, n2, a = 0.05)

h.prop(p1, p2, n1, n2, a = 0.05)

Arguments

p1	Proportion for group one (between 0 and 1).
p2	Proportion for group two (between 0 and 1).
n1	Sample size for group one.
n2	Sample size for group two.
a	Significance level used for confidence intervals. Defaults to 0.05.

Details

$h = 2 \arcsin \sqrt{p_1} - 2 \arcsin \sqrt{p_2}$

where $p_1$ and $p_2$ are proportions for groups 1 and 2, respectively.

Using a simple large-sample approximation (via the delta method), the standard error of $h$ can be taken as:

$\mathrm{SE}(h) \approx \sqrt{1 / n_1 + 1 / n_2}$

,

which leads to a $(1 - \alpha)$ confidence interval for $h$:

$h \pm z_{1 - \alpha/2} \, \mathrm{SE}(h).$

This effect size is commonly recommended for differences in proportions (Cohen, 1988) and is particularly useful for power analysis and meta-analysis when working directly with proportions.

Value

A list containing Cohen's $h$ effect size and related statistics:

• ‘h' – Cohen’s h.

• 'hlow', 'hhigh' – lower and upper confidence interval limits.

• 'h_lower_limit', 'h_upper_limit' – snake_case aliases for the confidence limits.

• 'p1', 'p2' – input proportions for each group.

• 'n1', 'n2' – sample sizes for each group, with snake_case aliases 'sample_size_1', 'sample_size_2'.

• 'z', 'p' – z statistic and p value for the difference in proportions using a pooled-proportion standard error.

• 'z_value', 'p_value' – snake_case aliases for the z statistic and p value.

• 'estimate' – APA-style formatted string for Cohen's h and its confidence interval.

• 'statistic' – APA-style formatted string for the z test of the difference in proportions.

Examples

h_prop(p1 = .25, p2 = .35, n1 = 100, n2 = 100, a = .05)

---

Independent-Samples t-Test Example Data

Description

Example data for an independent-samples t-test examining whether a hypnotism intervention affects recall accuracy after witnessing a crime. Designed for use with functions such as d_ind_t, d_ind_t_t, and delta_ind_t.

Usage

data(indt_data)

Format

A data frame with 2 variables:

correctq

Numeric recall score/accuracy.

group

Factor indicating condition with levels "control" and "hypnotism".

---

Mixed Two-Way ANOVA Example Data

Description

Example data for a mixed two-way ANOVA examining whether instructions to ignore backward strength in free association (BSG) reduce overestimation in response probabilities. Participants provided estimates of how many out of 100 people would say a given response to a target word ("Family Feud"-style), under low vs. high BSG conditions, after receiving either regular or debiasing instructions.

Usage

data(mix2_data)

Format

A data frame with 3 variables:

group

Factor indicating instruction condition with levels "Regular JAM Task" and "Debiasing JAM task".

bsglo

Numeric. Estimated response percentage in the Low BSG condition.

bsghi

Numeric. Estimated response percentage in the High BSG condition.

---

Odds Ratio from 2x2 Table

Description

This function displays odds ratios and their normal confidence intervals. This statistic is calculated as (level 1.1/level 1.2) / (level 2.1/level 2.2), which can be considered the odds of level 1.1 given level1 overall versus level2.1 given level2 overall.

Usage

odds_ratio(n11, n12, n21, n22, a = 0.05)

odds(n11, n12, n21, n22, a = 0.05)

Arguments

n11	sample size for level 1.1
n12	sample size for level 1.2
n21	sample size for level 2.1
n22	sample size for level 2.2
a	significance level

Details

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'odds_ratio()' to follow modern R style guidelines. The original name 'odds()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'odds', 'olow', 'ohigh', 'se') and newer snake_case aliases (e.g., 'odds_value', 'odds_lower_limit', 'odds_upper_limit', 'standard_error'). New code should prefer 'odds_ratio()' and the snake_case output names, but existing code using the older names will continue to work.

Value

odds

odds ratio statistic (legacy name; see also 'odds_value')

olow

lower-level confidence interval of odds ratio (legacy name; see also 'odds_lower_limit')

ohigh

upper-level confidence interval of odds ratio (legacy name; see also 'odds_upper_limit')

se

standard error (legacy name; see also 'standard_error')

odds_value

odds ratio statistic (snake_case alias of 'odds')

odds_lower_limit

lower-level confidence interval of odds ratio (alias of 'olow')

odds_upper_limit

upper-level confidence interval of odds ratio (alias of 'ohigh')

standard_error

standard error (alias of 'se')

Examples

# A health psychologist was interested in the rates of anxiety in
# first generation and regular college students. They polled campus
# and found the following data:

  # |              | First Generation | Regular |
  # |--------------|------------------|---------|
  # | Low Anxiety  | 10               | 50      |
  # | High Anxiety | 20               | 15      |

# What are the odds for the first generation students to have anxiety?

odds_ratio(n11 = 10, n12 = 50, n21 = 20, n22 = 15, a = .05)

# Backwards-compatible wrapper (deprecated name)
odds(n11 = 10, n12 = 50, n21 = 20, n22 = 15, a = .05)

---

$\omega^2$ for ANOVA from $F$

Description

This function displays $\omega^2$ from ANOVA analyses and its non-central confidence interval based on the $F$ distribution. These values are calculated directly from F statistics and can be used for between subjects and repeated measures designs. Remember if you have two or more IVs, these values are partial omega squared.

Usage

omega_f(dfm, dfe, f_value, n, a = 0.05, Fvalue)

omega.F(dfm, dfe, Fvalue, n, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
f_value	F statistic
n	full sample size
a	significance level
Fvalue	Backward-compatible argument for the F statistic (deprecated; use 'f_value' instead). If supplied, it overrides 'f_value'. Included for users of the legacy 'omega.F()'.

Details

Omega squared or partial omega squared is calculated by subtracting one from the $F$-statistic and multiplying it by degrees of freedom of the model. This is divided by the same value after adding the number of valid responses. This value will be omega squared for one-way ANOVA designs, and will be partial omega squared for multi-way ANOVA designs (i.e. with more than one IV).

$\omega^2 = \frac{df_m (F - 1)}{df_m (F - 1) + n}$

Learn more on our example page.

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'omega_f()' to follow modern R style guidelines. The original dotted version 'omega.F()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'omega', 'omegalow', 'omegahigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'omega_value', 'omega_lower_limit', 'omega_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'omega_f()' and the snake_case output names, but existing code using the older names will continue to work.

Value

omega

$\omega^2$ effect size (legacy name; see also 'omega_value')

omegalow

lower-level confidence interval of $\omega^2$ (legacy name; see also 'omega_lower_limit')

omegahigh

upper-level confidence interval of $\omega^2$ (legacy name; see also 'omega_upper_limit')

dfm

degrees of freedom for the model/IV/between (legacy name; see also 'df_model')

dfe

degrees of freedom for the error/residual/within (legacy name; see also 'df_error')

F

$F$-statistic (legacy name; see also 'f_value')

p

p-value (legacy name; see also 'p_value')

estimate

the $\omega^2$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

omega_value

$\omega^2$ effect size (snake_case alias of 'omega')

omega_lower_limit

lower-level confidence interval of $\omega^2$ (alias of 'omegalow')

omega_upper_limit

upper-level confidence interval of $\omega^2$ (alias of 'omegahigh')

df_model

degrees of freedom for the model/IV/between (alias of 'dfm')

df_error

degrees of freedom for the error/residual/within (alias of 'dfe')

f_value

$F$-statistic (alias of 'F')

p_value

p-value (alias of 'p')

Examples

# The following example is derived from
# the "bn1_data" dataset, included in the MOTE library.

# A health psychologist recorded the number of close inter-personal
# attachments of 45-year-olds who were in excellent, fair, or poor
# health. People in the Excellent Health group had 4, 3, 2, and 3
# close attachments; people in the Fair Health group had 3, 5,
# and 8 close attachments; and people in the Poor Health group
# had 3, 1, 0, and 2 close attachments.

anova_model <- lm(formula = friends ~ group, data = bn1_data)
summary.aov(anova_model)

omega_f(dfm = 2, dfe = 8,
        f_value = 5.134, n = 11, a = .05)

# Backwards-compatible dotted name (deprecated)
omega.F(dfm = 2, dfe = 8,
        Fvalue = 5.134, n = 11, a = .05)

---

omega^2 for One-Way and Multi-Way ANOVA from F

Description

This function displays $\omega^2$ from ANOVA analyses and its non-central confidence interval based on the $F$ distribution. This formula works for one way and multi way designs with careful focus on which error term you are using for the calculation.

Usage

omega_full_ss(dfm, dfe, msm, mse, sst, a = 0.05)

omega.full.SS(dfm, dfe, msm, mse, sst, a = 0.05)

Arguments

dfm	degrees of freedom for the model/IV/between
dfe	degrees of freedom for the error/residual/within
msm	mean square for the model/IV/between
mse	mean square for the error/residual/within
sst	sum of squares total
a	significance level

Details

Omega squared is calculated by deducting the mean square of the error from the mean square of the model and multiplying by the degrees of freedom for the model. This is divided by the sum of the sum of squares total and the mean square of the error.

$\omega^2 = \frac{df_m (ms_m - ms_e)}{SS_T + ms_e}$

Learn more on our example page.

**Note on function and output names:** This effect size is now implemented with the snake_case function name 'omega_full_ss()' to follow modern R style guidelines. The original dotted version 'omega.full.SS()' is still available as a wrapper for backward compatibility, and both functions return the same list. The returned object includes both the original element names (e.g., 'omega', 'omegalow', 'omegahigh', 'dfm', 'dfe', 'F', 'p', 'estimate', 'statistic') and newer snake_case aliases (e.g., 'omega_value', 'omega_lower_limit', 'omega_upper_limit', 'df_model', 'df_error', 'f_value', 'p_value'). New code should prefer 'omega_full_ss()' and the snake_case output names, but existing code using the older names will continue to work.

Value

omega

$\omega^2$ effect size (legacy name; see also 'omega_value')

omegalow

lower-level confidence interval of $\omega^2$ (legacy name; see also 'omega_lower_limit')

omegahigh

upper-level confidence interval of $\omega^2$ (legacy name; see also 'omega_upper_limit')

dfm

degrees of freedom for the model/IV/between (legacy name; see also 'df_model')

dfe

degrees of freedom for the error/residual/within (legacy name; see also 'df_error')

F

$F$-statistic (legacy name; see also 'f_value')

p

p-value (legacy name; see also 'p_value')

estimate

the $\omega^2$ statistic and confidence interval in APA style for markdown printing

statistic

the $F$-statistic in APA style for markdown printing

omega_value

$\omega^2$ effect size (snake_case alias of 'omega')

omega_lower_limit

lower-level confidence interval of $\omega^2$ (alias of 'omegalow')

omega_upper_limit

upper-level confidence interval of $\omega^2$ (alias of 'omegahigh')

df_model

degrees of freedom for the model/IV/between (alias of 'dfm')

df_error

degrees of freedom for the error/residual/within (alias of 'dfe')

f_value

$F$-statistic (alias of 'F')

p_value

p-value (alias of 'p')

Examples

# The following example is derived from the "bn1_data"
# dataset, included in the MOTE library.

# A health psychologist recorded the number of close inter-personal
# attachments of 45-year-olds who were in excellent, fair, or poor
# health. People in the Excellent Health group had 4, 3, 2, and 3
# close attachments; people in the Fair Health group had 3, 5,
# and 8 close attachments; and people in the Poor Health group
# had 3, 1, 0, and 2 close attachments.

anova_model <- lm(formula = friends ~ group, data = bn1_data)
summary.aov(anova_model)

omega_full_ss(dfm = 2, dfe = 8,
              msm = 12.621, mse = 2.548,
              sst = (25.54 + 19.67), a = .05)

# Backwards-compatible dotted name (deprecated)
omega.full.SS(dfm = 2, dfe = 8,
              msm = 12.621, mse = 2.548,
              sst = (25.54 + 19.67), a = .05)

---