Standardized moderated mediation

What is standardized?

For a two-condition design, define $D M=M_2-M_1$ and $D Y=Y_2-Y_1$. PrepareData() centers the mediator average component, but does not center these differences. Standardization divides a difference by its marginal model-implied SD, preserving its mean condition effect.

Write $s_M=SD(D M)$ and $s_Y=SD(D Y)$. Without moderation,

\[ a^*=a/s_M,\qquad b^*=b s_M/s_Y,\qquad c'^*=c'/s_Y, \qquad IE^*=ab/s_Y. \]

The condition contrast retains its original unit. In a serial chain, the intermediate mediator SDs cancel. Thus every complete indirect effect, its contrast with another complete indirect effect, the total indirect effect and the total condition effect use the outcome-difference SD.

These are marginal SDs implied by the fitted model, not residual SDs, group-specific SDs or SDs conditional on a selected moderator value. The same endpoint scales apply at every moderator value within a fit.

Continuous moderators: two different quantities

For a centered moderator $w_c=w-\bar w$, suppose the conditional paths are $a(w_c)=a+a_w w_c$ and $b(w_c)=b+b_w w_c$. Then

\[ IE^*(w_c)=\frac{(a+a_w w_c)(b+b_w w_c)}{s_Y}, \qquad b^*(w_c)=(b+b_w w_c)\frac{s_M}{s_Y}. \]

The standardized interaction coefficient is instead

\[ b_w^*=b_w\frac{s_M s_W}{s_Y},\qquad a_w^*=a_w\frac{s_W}{s_M}. \]

moderation_std$mod_coeff reports interaction coefficients per SD of a continuous moderator. Conditional tables and curves evaluate effects at fixed numerical moderator values in the original units. To recover a conditional slope from fully standardized coefficients, use $b^*(w_c)=b^*+b_w^*(w_c/s_W)$. The scale of an interaction is the product of its component scale factors, not the SD of the observed product column.

Fit a reproducible example

The following simulated data have a moderator with a nonzero mean and an SD different from one. The mediator averages and differences are generated separately to make the two-condition structure explicit.

set.seed(918)
n <- 350
w <- rnorm(n, 3, 2)
avg1 <- rnorm(n)
avg2 <- rnorm(n)
dm1 <- 2 + 0.8 * w + rnorm(n)
dm2 <- 1 + 0.4 * w + 0.3 * dm1 + rnorm(n)
dy <- 0.5 + 0.6 * w + 0.5 * dm1 + 0.7 * dm2 +
      0.2 * dm1 * w + 0.3 * avg1 + rnorm(n)
y1 <- rnorm(n)
dat <- data.frame(M11 = avg1 - dm1/2, M12 = avg1 + dm1/2,
                  M21 = avg2 - dm2/2, M22 = avg2 + dm2/2,
                  Y1 = y1, Y2 = y1 + dy, W = w)

result <- wsMed(
  data = dat, M_C1 = c("M11", "M21"), M_C2 = c("M12", "M22"),
  Y_C1 = "Y1", Y_C2 = "Y2", form = "P",
  W = "W", W_type = "continuous", MP = c("b1", "d1"),
  Na = "DE", ci_method = "mc", R = 2000, seed = 2026,
  fixed.x = FALSE, standardized = TRUE, verbose = FALSE
)
head(result$moderation_std$beta_coef)
#>                Path Mediators Level W_value  Estimate         SE 2.5.%CI.Lo 97.5.%CI.Up Sig
#> 1 indirect_effect_1         1 -1 SD   1.264 0.3659377 0.03230506  0.3050204   0.4302605   *
#> 2 indirect_effect_1         1  0 SD   3.233 0.8565050 0.05060222  0.7604226   0.9587515   *
#> 3 indirect_effect_1         1 +1 SD   5.202 1.5525336 0.07945743  1.4019444   1.7112131   *
#> 4 indirect_effect_2         2 -1 SD   1.264 0.2814658 0.02391658  0.2382061   0.3301252   *
#> 5 indirect_effect_2         2  0 SD   3.233 0.4522848 0.03551574  0.3877237   0.5277429   *
#> 6 indirect_effect_2         2 +1 SD   5.202 0.6231038 0.04872421  0.5335590   0.7248074   *
result$moderation_std$conditional_overall
#>           Effect Level W_value  Estimate         SE 2.5.%CI.Lo 97.5.%CI.Up Sig
#> 1 total_indirect -1 SD   1.264 0.6474035 0.04403820  0.5686612   0.7361678   *
#> 3 total_indirect  0 SD   3.233 1.3087898 0.06938467  1.1812746   1.4488353   *
#> 5 total_indirect +1 SD   5.202 2.1756374 0.10450931  1.9806249   2.3869356   *
#> 2   total_effect -1 SD   1.264 0.8359648 0.04061028  0.7614711   0.9227852   *
#> 4   total_effect  0 SD   3.233 1.6487966 0.06495734  1.5313773   1.7897487   *
#> 6   total_effect +1 SD   5.202 2.6670898 0.09830545  2.4921740   2.8805311   *
result$moderation_std$mod_coeff
#>     Path BaseCoef W_dummy     Estimate         SE   2.5%CI.Lo 97.5%CI.Up Sig
#> 1 bw1_W1       b1       W  0.120939967 0.00767366  0.10589394 0.13604213   *
#> 2 dw1_W1       d1       W -0.001567994 0.00778167 -0.01762458 0.01290788    
#> 3 cpw_W1       cp       W  0.151229035 0.02707509  0.09963155 0.20281562   *
#> 4 aw1_W1       a1       W  0.847008355 0.01539922  0.81451474 0.87424607   *
#> 5 aw2_W1       a2       W  0.811031030 0.01854189  0.77050108 0.84491935   *

MP requests interactions and focal paths. Moderator main effects already included in the fitted regressions also enter conditional intercepts, even when a1 or cp is absent from MP. The total indirect effect includes both mediator paths in this example, not just the path with a requested interaction.

The original result$moderation remains available. Without a moderator, standardized = TRUE supplies parameter tables such as result$mc$std_mc, but there is no conditional moderation table.

plot_moderation_curve(result, "total_indirect", standardized = TRUE)

plot of chunk plot-continuous

The horizontal axis retains raw moderator units. Shading marks stored grid values whose pointwise confidence interval excludes zero; boundary labels report raw W values. These are grid approximations, not exact analytical Johnson–Neyman boundaries or simultaneous confidence regions.

A check of the point estimates

For this complete-data MC fit, the standardized overall effects must equal the raw effects divided by the fitted marginal outcome SD:

sy <- sqrt(lavaan::fitted(result$mc$fit)$cov["Ydiff", "Ydiff"])
stopifnot(isTRUE(all.equal(
  result$moderation_std$conditional_overall$Estimate,
  result$moderation$conditional_overall$Estimate / sy,
  tolerance = 1e-7
)))
stopifnot(identical(result$moderation_std$theta_curve$W_raw,
                    result$moderation$theta_curve$W_raw))

This identity concerns point estimates. It does not justify dividing raw confidence limits by the fitted SD.

Categorical moderators

Categorical moderators retain their 0/1 dummy coding. There is no SD-of-dummy multiplier, and all groups share the same marginal endpoint scales. The factor levels below make the reference category explicit.

dat$Group <- cut(dat$W, c(-Inf, 2, 4, Inf), labels = c("A", "B", "C"))
levels(dat$Group)
#> [1] "A" "B" "C"
group_result <- wsMed(
  data = dat, M_C1 = c("M11", "M21"), M_C2 = c("M12", "M22"),
  Y_C1 = "Y1", Y_C2 = "Y2", form = "P",
  W = "Group", W_type = "categorical", MP = c("b1", "d1"),
  Na = "DE", ci_method = "mc", R = 2000, seed = 2026,
  fixed.x = FALSE, standardized = TRUE, verbose = FALSE
)
group_result$moderation_std$conditional_IE
#>           IE Group  Estimate         SE 2.5%CI.Lo 97.5%CI.Up Sig
#> 1 indirect_1     A 0.4317985 0.03949438 0.3609500  0.5155264   *
#> 2 indirect_1     B 0.9030258 0.10580085 0.7002119  1.1168575   *
#> 3 indirect_1     C 2.9031442 0.15930952 2.6070075  3.2351171   *
#> 4 indirect_2     A 0.4146559 0.03831356 0.3477088  0.4974758   *
#> 5 indirect_2     B 0.6610705 0.04894680 0.5725422  0.7642088   *
#> 6 indirect_2     C 0.9882944 0.06686062 0.8638378  1.1245713   *
group_result$moderation_std$IE_contrasts
#>           IE Contrast  Estimate         SE 2.5%CI.Lo 97.5%CI.Up Sig
#> 1 indirect_1    B - A 0.4712273 0.09552160 0.2848670  0.6589287   *
#> 2 indirect_1    C - A 2.4713457 0.14104375 2.2040159  2.7584021   *
#> 3 indirect_1    C - B 2.0001185 0.16125826 1.6990363  2.3274478   *
#> 4 indirect_2    B - A 0.2464146 0.03401355 0.1805038  0.3130077   *
#> 5 indirect_2    C - A 0.5736385 0.04630917 0.4836041  0.6680084   *
#> 6 indirect_2    C - B 0.3272238 0.03533971 0.2618490  0.4011361   *
group_result$moderation_std$conditional_overall
#>           Effect Group  Estimate         SE 2.5%CI.Lo 97.5%CI.Up Sig
#> 1 total_indirect     A 0.8464544 0.06061017 0.7347466  0.9787931   *
#> 3 total_indirect     B 1.5640963 0.12137367 1.3393435  1.8137175   *
#> 5 total_indirect     C 3.8914386 0.18295879 3.5709714  4.2775923   *
#> 2   total_effect     A 0.8262732 0.05725262 0.7252731  0.9507193   *
#> 4   total_effect     B 1.6669270 0.07365657 1.5298071  1.8200928   *
#> 6   total_effect     C 3.0760606 0.10983884 2.8734913  3.3155557   *

Use these group tables for categorical moderators. The continuous-moderator curve example above should not be interpreted as interpolating between groups.

Convert a fitted result without refitting

standardize_moderation() uses the joint MC or bootstrap draws stored in a wsMed object with a moderator. It returns the standardized moderation list:

converted <- standardize_moderation(result)
stopifnot(isTRUE(all.equal(converted, result$moderation_std)))
result$moderation_std <- converted
attr(converted, "standardization")
#> $effect_scale
#> [1] "marginal model-implied endpoint scales; IE / SD(Ydiff)"
#> 
#> $moderator_units
#> [1] "raw; fixed numerical probes; dummy coding unchanged"
#> 
#> $mod_coeff_units
#> [1] "per SD of continuous W; per 0/1 dummy contrast"
#> 
#> $interval
#> [1] "percentile"
#> 
#> $engine
#> [1] "mc"
#> 
#> $fixed.x
#> [1] FALSE
#> 
#> $MI_reference
#> NULL
#> 
#> $diagnostics
#> $diagnostics$requested
#> [1] 2000
#> 
#> $diagnostics$valid
#> [1] 2000
#> 
#> $diagnostics$invalid
#> [1] 0
#> 
#> $diagnostics$invalid_indices
#> integer(0)
#> 
#> $diagnostics$reasons
#> < table of extent 0 >

A result without the required stored draws cannot be converted by this helper.

Confidence intervals, bootstrap and both engines

Each valid joint draw supplies both regression coefficients and model-implied SDs. Transform the effect using those draw-specific SDs, then summarize the transformed draws. For continuous moderators, the selected raw values (for example the reference mean and mean plus/minus one reference SD) remain fixed across draws. If standardized coefficients are used, evaluate them at $w_c/s_W^{(r)}$ in draw $r$, not at one fixed standardized moderator value.

Point estimates evaluate the effect at the fitted parameters, or at pooled primitive parameters for MI. They are not averages of nonlinear draw effects. Invalid joint draws are excluded with diagnostics; inspect the standardization attribute for their counts, indices and reasons.

All conditional tables use percentile intervals, for both MC and bootstrap. boot_ci_type controls parameter-table intervals; it does not change conditional intervals. In standardized parameter tables, "bc" and "bca.simple" use bias correction with zero acceleration, not a full acceleration-adjusted BCa calculation. Standardized bootstrap p-values require at least 1000 valid replicates; that is not a minimum for conditional intervals.

The following longer bootstrap example is not run during vignette rebuilding:

both_result <- wsMed(
  data = dat, M_C1 = c("M11", "M21"), M_C2 = c("M12", "M22"),
  Y_C1 = "Y1", Y_C2 = "Y2", form = "P",
  W = "W", W_type = "continuous", MP = c("b1", "d1"),
  Na = "DE", ci_method = "both", R = 5000, bootstrap = 2000,
  seed = 2026, iseed = 2026, boot_ci_type = "perc",
  standardized = TRUE
)
both_result$moderation_std$mc$conditional_overall
both_result$moderation_std$boot$conditional_overall
plot_moderation_curve(both_result, "total_indirect",
                      standardized = TRUE, engine = "boot")

For a single engine ("mc" or "bootstrap"), tables are directly inside moderation_std. For "both", they are nested under $mc and $boot; the plotting engine argument selects between them.

Missing data and reporting

FIML and MI use the same effect definition. With fixed.x = FALSE, estimated external moments participate in the joint transformation. With fixed.x = TRUE, transformations condition on fitted external moments. MI supports MC inference and evaluates effects at pooled primitive parameters; its probing and centering reference is the first completed dataset. With MI and fixed.x = TRUE, external moments from that reference are held fixed. This does not introduce population-relative moderator probing or a new imputation procedure.

Report the condition difference direction, model and moderated paths, moderator coding/reference and raw probe values, marginal SD definition, missing-data method, fixed.x, inference engine, number of valid draws and interval type. Distinguish standardized interaction coefficients per SD of W from conditional indirect effects at particular raw W values.

For serial and sparse models, see the custom-model tutorial. Their complete indirect effects follow the same endpoint scaling rule.