Given the hypothetical population effect size and the study sample size, the
function retrospective() performs a retrospective design analysis for
Pearson's correlation test between two variables or t-test comparing
group means (Cohen's d). According to the defined alternative
hypothesis and the significance level, inferential risks (i.e., Power level,
Type M error, and Type S error) are computed together with the critical
effect value (i.e., the minimum absolute effect size value that would result
significant).
retrospective( effect_size, sample_n1, sample_n2 = NULL, test_method = c("pearson", "two_sample", "welch", "paired", "one_sample"), alternative = c("two_sided", "less", "greater"), sig_level = 0.05, ratio_sd = 1, B = 10000, tl = -Inf, tu = Inf, B_effect = 1000, display_message = TRUE )
| effect_size | a numeric value or function (see Details) indicating the hypothetical population effect size. |
|---|---|
| sample_n1 | a numeric value indicating the sample size of the first group. |
| sample_n2 | a numeric value indicating the sample size of the second
group. This argument is required when |
| test_method | a character string specifying the test type, must be one of
|
| alternative | a character string specifying the alternative hypothesis,
must be one of |
| sig_level | a numeric value indicating the significance level on which the alternative hypothesis is evaluated. |
| ratio_sd | a numeric value indicating the ratio between the standard deviation in the first group and in the second group. This argument is needed in the case of Welch's t-test. |
| B | a numeric value indicating the number of iterations. Increase the number of iterations to obtain more stable results. |
| tl | optional value indicating the lower truncation point if
|
| tu | optional value indicating the upper truncation point if
|
| B_effect | a numeric value indicating the number of sampled effects
if |
| display_message | a logical variable indicating whether to display or not
the progress bar. Not that this applies only when |
A list with class "design_analysis" containing the following components:
a character string indicating the type of design analysis: "retrospective".
a list with all the arguments passed to the function and the raw function call.
a list with all the information regarding the
considered hypothetical population effect size. The list includes:
effect_type indicating the type of effect; effect_function
indicating the function from which effect are sampled or the string
"single_value" if a single value was provided; effect_summary
summary of the sampled effects; effect_samples vector with the
sampled effects (or unique value in the case of a single value). if
relevant tl and tu specifying the lower upper truncation
point respectively.
a list with all the information regarding the test
performed. The list includes: test_method character sting
indicating the test method (i.e., "pearson", "one_sample", "paired",
"two_sample", or "welch"); sample size (sample_n1 and if relevant
sample_n2), alternative hypothesis (alternative),
significance level (sig_level) and degrees of freedom (df)
of the statistical test; critical_effect the minimum absolute
effect value that would result significant. Note that
critical_effect in the case of alternative = "two_sided" is
the absolute value and both positive and negative values should be
considered.
a data frame with the results of the design
analysis. Columns names are power, typeM, and typeS.
Conduct a retrospective design analysis to evaluate inferential risks
according to study design. A general overview is provided in the
vignette("retrospective").
Population effect size
The hypothetical population effect size (effect_size) can be set to a
single value or a function that allows sampling values from a given
distribution. The function has to be defined as function(n)
my_function(n, ...), with only one single argument n representing
the number of sampled values (e.g., function(n) rnorm(n, mean = 0, sd
= 1); function(n) sample(c(.1,.3,.5), n, replace = TRUE)). This
allows users to define hypothetical effect size distribution according to
their needs.
Argument B_effect allows defining the number of sampled effects.
Users can access sampled effects in the effect_info list included in
the output to evaluate if the sample is representative of their
specification. Increase the number to obtain more accurate results but it
will require more computational time (default is 1000). To avoid long
computational times, we suggest adjusting B when using a function to
define the hypothetical population effect size.
Optional arguments tl and tu allow truncating the sampling
distribution specifying the lower truncation point and upper truncation
point respectively. Note that if effect_type = "correlation",
distribution is automatically truncated between -1 and 1.
Test methods
The function retrospective() performs a retrospective design analysis
considering correlations between two variables or comparisons between group
means.
In the case of a correlation, only Pearson's correlation between two
variables is available, whereas Kendall's tau and Spearman's
rho are not implemented. The test_method argument has to be
set to "pearson" (default) and the effect_size argument is
used to define the hypothetical population effect size in terms of Pearson's
correlation coefficient (\(\rho\)). The sample_n2 argument is
ignored.
In the case of a comparison between group means, the effect_size
argument is used to define the hypothetical population effect size in terms
of Cohen's d and the available t-tests are selected specifying
the argument test_method. For independent two-sample t-test,
use "two_sample" and indicate the sample size of the second group
(sample_n2). For Welch's t-test, use "welch" and
indicate and indicate the sample size of the second group (sample_n2)
and the ratio between the standard deviation in the first group and in the
second group (ratio_sd). For dependent t-test for paired
samples, use "paired" (sample_n1 and sample_n2 have to
be equal). For one-sample t-test, use "one_sample"
(sample_n2 has to be NULL).
Study design
Study design can be further defined according to statistical test
directionality and required \(\alpha\)-level using the arguments
alternative and sig_level respectively.
Altoè, G., Bertoldo, G., Zandonella Callegher, C., Toffalini, E., Calcagnì, A., Finos, L., & Pastore, M. (2020). Enhancing Statistical Inference in Psychological Research via Prospective and Retrospective Design Analysis. Frontiers in Psychology, 10. https://doi.org/10.3389/fpsyg.2019.02893
Bertoldo, G., Altoè, G., & Zandonella Callegher, C. (2020). Designing Studies and Evaluating Research Results: Type M and Type S Errors for Pearson Correlation Coefficient. Retrieved from https://psyarxiv.com/q9f86/
Gelman, A., & Carlin, J. (2014). Beyond Power Calculations: Assessing Type S (Sign) and Type M (Magnitude) Errors. Perspectives on Psychological Science, 9(6), 641–651. https://doi.org/10.1177/1745691614551642
# Pearson's correlation retrospective(effect_size = .3, sample_n1 = 25, test_method = "pearson")#> #> Design Analysis #> #> Hypothesized effect: rho = 0.3 #> #> Study characteristics: #> test_method sample_n1 sample_n2 alternative sig_level df #> pearson 25 NULL two_sided 0.05 23 #> #> Inferential risks: #> power typeM typeS #> 0.313 1.678 0.001 #> #> Critical value(s): rho = ± 0.396# Two-sample t-test retrospective(effect_size = .3, sample_n1 = 25, sample_n2 = 35, test_method = "two_sample")#> #> Design Analysis #> #> Hypothesized effect: cohen_d = 0.3 #> #> Study characteristics: #> test_method sample_n1 sample_n2 alternative sig_level df #> two_sample 25 35 two_sided 0.05 58 #> #> Inferential risks: #> power typeM typeS #> 0.205 2.243 0.003 #> #> Critical value(s): cohen_d = ± 0.524# Welch t-test retrospective(effect_size = .3, sample_n1 = 25, sample_n2 = 35, test_method = "welch", ratio_sd = 1.5)#> #> Design Analysis #> #> Hypothesized effect: cohen_d = 0.3 #> #> Study characteristics: #> test_method sample_n1 sample_n2 alternative sig_level df #> welch 25 35 two_sided 0.05 38.89 #> #> Inferential risks: #> power typeM typeS #> 0.183 2.345 0.007 #> #> Critical value(s): cohen_d = ± 0.546# Paired t-test retrospective(effect_size = .3, sample_n1 = 25, sample_n2 = 25, test_method = "paired")#> #> Design Analysis #> #> Hypothesized effect: cohen_d = 0.3 #> #> Study characteristics: #> test_method sample_n1 sample_n2 alternative sig_level df #> paired 25 25 two_sided 0.05 24 #> #> Inferential risks: #> power typeM typeS #> 0.298 1.8 0 #> #> Critical value(s): cohen_d = ± 0.413# One-sample t-test retrospective(effect_size = .3, sample_n1 = 25, sample_n2 = NULL, test_method = "one_sample")#> #> Design Analysis #> #> Hypothesized effect: cohen_d = 0.3 #> #> Study characteristics: #> test_method sample_n1 sample_n2 alternative sig_level df #> one_sample 25 NULL two_sided 0.05 24 #> #> Inferential risks: #> power typeM typeS #> 0.303 1.873 0.001 #> #> Critical value(s): cohen_d = ± 0.413# \donttest{ # Define effect_size using functions (long computational times) # Remember to adjust B retrospective(effect_size = function(n) rnorm(n, .3, .1), sample_n1 = 25, test_method = "pearson", tl = .15, B = 1e3)#>#>#> #> Design Analysis #> #> Hypothesized effect: rho ~ rnorm(n, 0.3, 0.1) [tl = 0.15 ; tu = 1 ] #> n_effect Min. 1st Qu. Median Mean 3rd Qu. Max. #> 1000 0.15 0.249 0.308 0.312 0.371 0.624 #> #> Study characteristics: #> test_method sample_n1 sample_n2 alternative sig_level df #> pearson 25 NULL two_sided 0.05 23 #> #> Inferential risks: #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> power 0.102 0.22575 0.329 0.355294 0.46375 0.936 #> typeM 1.025 1.39900 1.627 1.743234 1.97925 3.225 #> typeS 0.000 0.00000 0.000 0.004147 0.00400 0.071 #> #> Critical value(s): rho = ± 0.396retrospective(effect_size = function(n) rnorm(n, .3, .1), sample_n1 = 25, test_method = "one_sample", tl = .2, tu = .4, B = 1e3)#>#> #> Design Analysis #> #> Hypothesized effect: cohen_d ~ rnorm(n, 0.3, 0.1) [tl = 0.2 ; tu = 0.4 ] #> n_effect Min. 1st Qu. Median Mean 3rd Qu. Max. #> 1000 0.2 0.258 0.299 0.3 0.342 0.4 #> #> Study characteristics: #> test_method sample_n1 sample_n2 alternative sig_level df #> one_sample 25 NULL two_sided 0.05 24 #> #> Inferential risks: #> Min. 1st Qu. Median Mean 3rd Qu. Max. #> power 0.146 0.236 0.300 0.308001 0.37600 0.516 #> typeM 1.454 1.673 1.866 1.917553 2.11825 2.690 #> typeS 0.000 0.000 0.000 0.002266 0.00400 0.026 #> #> Critical value(s): cohen_d = ± 0.413# }