Author: Shu Fai Cheung
Date: 2023-09-09

# Conditional Effects by cond_effect()

#### Shu Fai Cheung

#### 2023-09-09

# Introduction

This vignette illustrates how to use `cond_effect()` from the `stdmod` package. More about this package can be found in `vignette("stdmod", package = "stdmod")` or at https://sfcheung.github.io/stdmod/.

# What `cond_effect()` Can Do

It can compute the conditional effect of a predictor (the focal variable) on an outcome variable (dependent variable) for selected levels of the moderator:

    #>   Level conscientiousness emotional_stability Effect  S.E.     t     p Sig
    #>    High             3.950                      0.012 0.117 0.107 0.915
    #>  Medium             3.343                      0.214 0.083 2.560 0.011 *
    #>     Low             2.736                      0.415 0.115 3.601 0.000 ***

It can also compute standardized conditional moderation effect of this predictor:

    #>   Level conscientiousness emotional_stability Effect  S.E.     t     p Sig
    #>    High             1.000                      0.007 0.063 0.107 0.915
    #>  Medium             0.000                      0.115 0.045 2.560 0.011 *
    #>     Low            -1.000                      0.223 0.062 3.601 0.000 ***

Nonparametric bootstrap percentile confidence interval can also be formed for standardized conditional effect using `cond_effect_boot()`.

`cond_effect()` is not designed to be a versatile tool. It is designed to be a function “good-enough” for common scenarios. Nevertheless, it can report some useful information along with the conditional effects, as demonstrated below.

# Major Arguments

## Regression Output, Predictor (`x`), and Moderator (`w`)

- `output`: The output of `lm()`, `std_selected()`, or `std_selected_boot()`, with at least one interaction term. Bootstrap estimates in `std_selected_boot()` will be ignored because bootstrapping will be done for each level again.

- `x`: The predictor (focal variable).

- `w`: The moderator.

These are the only required arguments. Just setting them can generate the graph:

    library(stdmod)
    data(sleep_emo_con)
    lm_out <- lm(sleep_duration ~ age + gender +
                                  emotional_stability * conscientiousness,
                                  sleep_emo_con)
    cond_out <- cond_effect(output = lm_out,
                            x = "emotional_stability",
                            w = "conscientiousness")
    cond_out
    #> The effects of emotional_stability on sleep_duration, conditional on conscientiousness:
    #>
    #>   Level conscientiousness emotional_stability Effect  S.E.     t     p Sig
    #>    High             3.950                      0.012 0.117 0.107 0.915
    #>  Medium             3.343                      0.214 0.083 2.560 0.011 *
    #>     Low             2.736                      0.415 0.115 3.601 0.000 ***
    #>
    #>
    #> The regression model:
    #>
    #>  sleep_duration ~ age + gender + emotional_stability * conscientiousness
    #>
    #> Interpreting the levels of conscientiousness:
    #>
    #>   Level conscientiousness % Below From Mean (in SD)
    #>    High             3.950   83.60              1.00
    #>  Medium             3.343   49.60              0.00
    #>     Low             2.736   16.60             -1.00
    #>
    #> - % Below: The percent of cases equal to or less than a level.
    #> - From Mean (in SD): Distance of a level from the mean, in standard
    #>   deviation (+ve above, -ve below).

By default, the print method of `cond_effect()` output prints the conditional effects, OLS standard errors, t statistics, p-values, and significant test results, along with other information such as the value of each level of the moderator, its distance from the mean, the percentage of cases equal to or less than this level. The regression model is also printed. If only the table of effects is needed, call `print()` and set `table_only` to `TRUE`:

    print(cond_out, table_only = TRUE)
    #>   Level conscientiousness emotional_stability Effect  S.E.     t     p Sig
    #>    High             3.950                      0.012 0.117 0.107 0.915
    #>  Medium             3.343                      0.214 0.083 2.560 0.011 *
    #>     Low             2.736                      0.415 0.115 3.601 0.000 ***

More options in printing the output can be found in the help page of `print.cond_effect()`.

# Levels of the Moderator

## Numeric Moderators

If the moderator is a numeric variable, then, by default, the conditional effects for three levels of the moderators will be used: one standard deviation (SD) to the mean (“Low”), mean (“Medium”), and one SD above mean (“High”).

Users can also use percentiles to define “Low”, “Medium”, and “High” by setting `w_method` to `"percentile"`. The default are 16th percentile, 50th percentile, and 84th percentile, which corresponds approximately to one SD below mean, mean, and one SD above mean, respectively, for a normal distribution.

    data(sleep_emo_con)
    lm_out <- lm(sleep_duration ~ age + gender +
                                  emotional_stability * conscientiousness,
                                  sleep_emo_con)
    cond_out <- cond_effect(output = lm_out,
                            x = "emotional_stability",
                            w = "conscientiousness",
                            w_method = "percentile")
    print(cond_out, title = FALSE, model = FALSE)
    #>   Level conscientiousness emotional_stability Effect  S.E.      t     p Sig
    #>    High             4.000                     -0.004 0.122 -0.034 0.973
    #>  Medium             3.400                      0.195 0.084  2.322 0.021 *
    #>     Low             2.700                      0.427 0.119  3.600 0.000 ***
    #>
    #> Interpreting the levels of conscientiousness:
    #>
    #>   Level conscientiousness % Below From Mean (in SD)
    #>    High             4.000   87.20              1.08
    #>  Medium             3.400   57.00              0.09
    #>     Low             2.700   16.60             -1.06
    #>
    #> - % Below: The percent of cases equal to or less than a level.
    #> - From Mean (in SD): Distance of a level from the mean, in standard
    #>   deviation (+ve above, -ve below).

Note that the empirical percentage of cases equal to or less than a level may not be exactly equal to that for the requested percentile if the number of cases is small and/or the number of unique values of the moderator is small.

## Categorical Moderators

If the moderator is a categorical variable (a string variable or a factor), then the conditional effect of the moderator for each value of this categorical moderator will be printed:

    set.seed(61452)
    sleep_emo_con$city <- sample(c("Alpha", "Beta", "Gamma"),
                                   nrow(sleep_emo_con), replace = TRUE)
    lm_cat <- lm(sleep_duration ~ age + gender + emotional_stability*city,
                                  sleep_emo_con)
    cond_out <- cond_effect(lm_cat,
                            x = "emotional_stability",
                            w = "city")
    print(cond_out, title = FALSE, model = FALSE)
    #>  Level  city emotional_stability Effect  S.E.     t     p Sig
    #>  Alpha Alpha                      0.408 0.135 3.027 0.003  **
    #>   Beta  Beta                      0.351 0.147 2.388 0.017  *
    #>  Gamma Gamma                      0.020 0.149 0.131 0.896

# Nonparametric Bootstrap Confidence Intervals

If one or more variables are standardized, the OLS confidence intervals are not appropriate (Cheung, Cheung, Lau, Hui, & Vong, 2022; Yuan & Chan, 2011). Users can call `cond_effect_boot()` to use nonparametric bootstrapping to form the percentile confidence interval for each conditional effect.

- `conf`: The level of confidence, expressed as a proportion. Default is .95, requesting a 95% confidence interval.
- `nboot` The number of bootstrap samples to drawn. Should be at least 2000 but 5000 is preferable.

To make the results reproducible, call `set.seed()` before calling `cond_effect_boot()`, as illustrated below.

    lm_out <- lm(sleep_duration ~ age + gender +
                                  emotional_stability * conscientiousness,
                                  sleep_emo_con)
    # Standardize all variables and do the moderated regression again
    # Use to_standardize as a shortcut to to_center and to_scale
    lm_std <- std_selected(lm_out,
                           to_standardize = ~ .)
    set.seed(897043)
    cond_std <- cond_effect_boot(output = lm_std,
                                 x = "emotional_stability",
                                 w = "conscientiousness",
                                 nboot = 2000)
    print(cond_std, model = FALSE, title = FALSE, level_info = FALSE)
    #>   Level conscientiousness emotional_stability Effect CI Lower CI Upper  S.E.
    #>    High             1.000                      0.007   -0.107    0.117 0.063
    #>  Medium             0.000                      0.115    0.029    0.202 0.045
    #>     Low            -1.000                      0.223    0.071    0.364 0.062
    #>      t     p Sig
    #>  0.107 0.915
    #>  2.560 0.011 *
    #>  3.601 0.000 ***
    #>
    #> [CI Lower, CI Upper] shows the 95% nonparametric bootstrap confidence
    #> interval(s) (based on 2000 bootstrap samples).
    #>
    #> Note:
    #>
    #> - The variable(s) sleep_duration, emotional_stability,
    #>   conscientiousness is/are standardized.
    #> - The conditional effects are the standardized effects of
    #>   emotional_stability on sleep_duration.

# Further Information

Please refer to the help page of `cond_effect()` and `cond_effect_boot()` for other options available, such as defining the number of SDs from mean to define “Low” and “High”, the percentiles to be used, or using parallel processing to speed up bootstrapping.

# Reference

Cheung, S. F., Cheung, S.-H., Lau, E. Y. Y., Hui, C. H., & Vong, W. N. (2022) Improving an old way to measure moderation effect in standardized units. Health Psychology, 41(7), 502-505. https://doi.org/10.1037/hea0001188.

Yuan, K.-H., & Chan, W. (2011). Biases and standard errors of standardized regression coefficients. Psychometrika, 76(4), 670-690. https://doi.org/10.1007/s11336-011-9224-6