Empirical Example
Ivan Jacob Agaloos Pesigan
Source:vignettes/empirical-example.Rmd
empirical-example.RmdIn this empirical example, we use data from a study by Elliot and others (2007) on the effects of an intervention on healthy dietary behavior mediated by knowledge of healthy dietary behavior. This data was used as an empirical example in Yuan and MacKinnon (2009) and Wu and Jia (2013).
Data
The data frame elliot2007(), which is included in the
manMCMedMiss package, has 354 cases and 3 variables:
-
x- Intervention group membership. -
m- Knowledge of healthy dietary behavior (knowledge post-intervention minus knowledge pre-intervention). -
y- Healthy dietary behavior (behavior post-intervention minus behavior pre-intervention).
| x | m | y |
|---|---|---|
| 0 | 1.0 | 1.666666 |
| 0 | 0.0 | 4.000000 |
| 0 | 0.0 | 0.000000 |
| 0 | -0.5 | -0.666667 |
| 0 | 0.5 | 1.333333 |
| 0 | 1.0 | 2.333334 |
Arguments
| Variables | Values | Notes |
|---|---|---|
| m | 100 | Number of imputations. |
| R | 20000 | Number of Monte Carlo replications. |
| B | 5000 | Number of bootstrap samples. |
| mplus_bin | “/opt/mplusdemo/mpdemo” | Path to Mplus binary. |
NOTE: If you are using
manmcmedmiss-rockerormanmcmedmiss.sifdescribed in the Containers article, setmplus_bin = "mpdemo".
Amputation
Generate sample data with missing values. The missing data mechanism is missing at random (MAR). The proportion of missing cases is 0.30.
set.seed(42)
data_missing <- AmputeData(
elliot2007,
mech = "MAR",
prop = 0.10
)
knitr::kable(head(data_missing, n = 10))| x | m | y |
|---|---|---|
| 0 | 1.000000 | 1.666666 |
| 0 | 0.000000 | 4.000000 |
| 0 | 0.000000 | 0.000000 |
| 0 | -0.500000 | -0.666667 |
| 0 | 0.500000 | 1.333333 |
| 0 | 1.000000 | 2.333334 |
| NA | 1.166667 | 3.333333 |
| NA | 0.833333 | NA |
| 0 | 1.666667 | 1.666667 |
| 0 | -0.333333 | 1.666666 |
Imputation
Perform multiple imputations
data_mi <- ImputeData(
data_missing,
m = m,
mplus_bin = mplus_bin
)Model Fitting
Maximum Likelihood
Parameters of the simple mediation model are estimated using full-information maximum likelihood to handle missing data.
fit_ml <- FitModelML(
data_missing,
mplus_bin = mplus_bin
)
fit_ml
#> $fit
#> free_parameters h0_loglikelihood h1_loglikelihood aic
#> 9.000000e+00 -1.225911e+03 -1.225911e+03 2.469821e+03
#> bic sabic chisq chisq_df
#> 2.504645e+03 2.476093e+03 8.237393e-06 0.000000e+00
#> chisq_p cfi tli rmsea
#> 0.000000e+00 9.999994e-01 1.000000e+00 0.000000e+00
#> rmsea_low rmsea_up rmsea_p srmr
#> 0.000000e+00 0.000000e+00 0.000000e+00 5.887405e-05
#> condition_number
#> 1.837067e-02
#>
#> $coef
#> m~1 y~1 x~1 m~x y~m y~x
#> 0.006934506 0.336487350 0.411909970 0.369397850 0.152114330 0.151127860
#> m~~m y~~y x~~x
#> 1.211989500 1.149110200 0.241468090
#>
#> $vcov
#> m~1 y~1 x~1 m~x y~m
#> m~1 6.200529e-03 4.010312e-05 -7.774377e-05 -6.518637e-03 6.355296e-05
#> y~1 4.010312e-05 6.012550e-03 -7.506271e-05 -7.453915e-05 1.085284e-04
#> x~1 -7.774377e-05 -7.506271e-05 7.515655e-04 1.438111e-04 -9.078051e-06
#> m~x -6.518637e-03 -7.453915e-05 1.438111e-04 1.649430e-02 3.655520e-05
#> y~m 6.355296e-05 1.085284e-04 -9.078051e-06 3.655520e-05 3.115320e-03
#> y~x -1.365620e-04 -6.342203e-03 1.720138e-04 8.861812e-05 -1.121932e-03
#> m~~m 1.191213e-04 1.111450e-05 -2.519574e-05 -2.088454e-04 -2.211516e-06
#> y~~y -9.056253e-06 4.214630e-05 -1.378773e-05 -3.796332e-05 -3.804997e-05
#> x~~x -5.274115e-06 -4.978534e-06 7.624901e-06 6.937207e-06 -1.102431e-06
#> y~x m~~m y~~y x~~x
#> m~1 -1.365620e-04 1.191213e-04 -9.056253e-06 -5.274115e-06
#> y~1 -6.342203e-03 1.111450e-05 4.214630e-05 -4.978534e-06
#> x~1 1.720138e-04 -2.519574e-05 -1.378773e-05 7.624901e-06
#> m~x 8.861812e-05 -2.088454e-04 -3.796332e-05 6.937207e-06
#> y~m -1.121932e-03 -2.211516e-06 -3.804997e-05 -1.102431e-06
#> y~x 1.622654e-02 -3.583622e-05 -7.859534e-05 1.124238e-05
#> m~~m -3.583622e-05 8.808493e-03 -4.183286e-06 -4.881001e-06
#> y~~y -7.859534e-05 -4.183286e-06 8.050239e-03 -1.407881e-06
#> x~~x 1.124238e-05 -4.881001e-06 -1.407881e-06 3.625790e-04
#>
#> $output
#> [1] "Mplus VERSION 8.8 DEMO (Linux)"
#> [2] "MUTHEN & MUTHEN"
#> [3] "09/18/2023 9:24 AM"
#> [4] ""
#> [5] "INPUT INSTRUCTIONS"
#> [6] ""
#> [7] " DATA:"
#> [8] " FILE = file8ff46918c33;"
#> [9] " VARIABLE:"
#> [10] " NAMES = X M Y;"
#> [11] " USEVARIABLES = X M Y;"
#> [12] " MISSING = ALL (-999);"
#> [13] " ANALYSIS:"
#> [14] " TYPE = GENERAL;"
#> [15] " ESTIMATOR = ML;"
#> [16] " MODEL:"
#> [17] " Y ON X;"
#> [18] " Y ON M (B);"
#> [19] " M ON X (A);"
#> [20] " X WITH X;"
#> [21] " SAVEDATA:"
#> [22] " RESULTS ARE file8ff6dd44a44;"
#> [23] " TECH3 IS file8ff10ac78d0;"
#> [24] ""
#> [25] ""
#> [26] ""
#> [27] ""
#> [28] "INPUT READING TERMINATED NORMALLY"
#> [29] ""
#> [30] ""
#> [31] ""
#> [32] ""
#> [33] "SUMMARY OF ANALYSIS"
#> [34] ""
#> [35] "Number of groups 1"
#> [36] "Number of observations 354"
#> [37] ""
#> [38] "Number of dependent variables 2"
#> [39] "Number of independent variables 1"
#> [40] "Number of continuous latent variables 0"
#> [41] ""
#> [42] "Observed dependent variables"
#> [43] ""
#> [44] " Continuous"
#> [45] " M Y"
#> [46] ""
#> [47] "Observed independent variables"
#> [48] " X"
#> [49] ""
#> [50] ""
#> [51] "Estimator ML"
#> [52] "Information matrix OBSERVED"
#> [53] "Maximum number of iterations 1000"
#> [54] "Convergence criterion 0.500D-04"
#> [55] "Maximum number of steepest descent iterations 20"
#> [56] "Maximum number of iterations for H1 2000"
#> [57] "Convergence criterion for H1 0.100D-03"
#> [58] ""
#> [59] "Input data file(s)"
#> [60] " file8ff46918c33"
#> [61] ""
#> [62] "Input data format FREE"
#> [63] ""
#> [64] ""
#> [65] "SUMMARY OF DATA"
#> [66] ""
#> [67] " Number of missing data patterns 7"
#> [68] ""
#> [69] ""
#> [70] "COVARIANCE COVERAGE OF DATA"
#> [71] ""
#> [72] "Minimum covariance coverage value 0.100"
#> [73] ""
#> [74] ""
#> [75] " PROPORTION OF DATA PRESENT"
#> [76] ""
#> [77] ""
#> [78] " Covariance Coverage"
#> [79] " M Y X"
#> [80] " ________ ________ ________"
#> [81] " M 0.946"
#> [82] " Y 0.890 0.929"
#> [83] " X 0.879 0.876 0.910"
#> [84] ""
#> [85] ""
#> [86] ""
#> [87] "UNIVARIATE SAMPLE STATISTICS"
#> [88] ""
#> [89] ""
#> [90] " UNIVARIATE HIGHER-ORDER MOMENT DESCRIPTIVE STATISTICS"
#> [91] ""
#> [92] " Variable/ Mean/ Skewness/ Minimum/ % with Percentiles"
#> [93] " Sample Size Variance Kurtosis Maximum Min/Max 20%/60% 40%/80% Median"
#> [94] ""
#> [95] " M 0.148 -0.290 -5.000 0.30% -0.667 0.000 0.000"
#> [96] " 335.000 1.244 2.398 4.000 0.30% 0.333 1.000"
#> [97] " Y 0.412 0.246 -3.000 0.30% -0.333 0.000 0.333"
#> [98] " 329.000 1.186 0.597 4.000 0.61% 0.667 1.333"
#> [99] " X 0.407 0.379 0.000 59.32% 0.000 0.000 0.000"
#> [100] " 322.000 0.241 -1.856 1.000 40.68% 1.000 1.000"
#> [101] ""
#> [102] ""
#> [103] "THE MODEL ESTIMATION TERMINATED NORMALLY"
#> [104] ""
#> [105] ""
#> [106] ""
#> [107] "MODEL FIT INFORMATION"
#> [108] ""
#> [109] "Number of Free Parameters 9"
#> [110] ""
#> [111] "Loglikelihood"
#> [112] ""
#> [113] " H0 Value -1225.911"
#> [114] " H1 Value -1225.911"
#> [115] ""
#> [116] "Information Criteria"
#> [117] ""
#> [118] " Akaike (AIC) 2469.821"
#> [119] " Bayesian (BIC) 2504.645"
#> [120] " Sample-Size Adjusted BIC 2476.093"
#> [121] " (n* = (n + 2) / 24)"
#> [122] ""
#> [123] "Chi-Square Test of Model Fit"
#> [124] ""
#> [125] " Value 0.000"
#> [126] " Degrees of Freedom 0"
#> [127] " P-Value 0.0000"
#> [128] ""
#> [129] "RMSEA (Root Mean Square Error Of Approximation)"
#> [130] ""
#> [131] " Estimate 0.000"
#> [132] " 90 Percent C.I. 0.000 0.000"
#> [133] " Probability RMSEA <= .05 0.000"
#> [134] ""
#> [135] "CFI/TLI"
#> [136] ""
#> [137] " CFI 1.000"
#> [138] " TLI 1.000"
#> [139] ""
#> [140] "Chi-Square Test of Model Fit for the Baseline Model"
#> [141] ""
#> [142] " Value 17.825"
#> [143] " Degrees of Freedom 3"
#> [144] " P-Value 0.0005"
#> [145] ""
#> [146] "SRMR (Standardized Root Mean Square Residual)"
#> [147] ""
#> [148] " Value 0.000"
#> [149] ""
#> [150] ""
#> [151] ""
#> [152] "MODEL RESULTS"
#> [153] ""
#> [154] " Two-Tailed"
#> [155] " Estimate S.E. Est./S.E. P-Value"
#> [156] ""
#> [157] " Y ON"
#> [158] " X 0.151 0.127 1.186 0.235"
#> [159] " M 0.152 0.056 2.725 0.006"
#> [160] ""
#> [161] " M ON"
#> [162] " X 0.369 0.128 2.876 0.004"
#> [163] ""
#> [164] " Means"
#> [165] " X 0.412 0.027 15.025 0.000"
#> [166] ""
#> [167] " Intercepts"
#> [168] " M 0.007 0.079 0.088 0.930"
#> [169] " Y 0.336 0.078 4.339 0.000"
#> [170] ""
#> [171] " Variances"
#> [172] " X 0.241 0.019 12.681 0.000"
#> [173] ""
#> [174] " Residual Variances"
#> [175] " M 1.212 0.094 12.914 0.000"
#> [176] " Y 1.149 0.090 12.807 0.000"
#> [177] ""
#> [178] ""
#> [179] "QUALITY OF NUMERICAL RESULTS"
#> [180] ""
#> [181] " Condition Number for the Information Matrix 0.184E-01"
#> [182] " (ratio of smallest to largest eigenvalue)"
#> [183] ""
#> [184] ""
#> [185] "RESULTS SAVING INFORMATION"
#> [186] ""
#> [187] " Order of data"
#> [188] ""
#> [189] " Parameter estimates"
#> [190] " (saved in order shown in Technical 1 output)"
#> [191] " Standard errors"
#> [192] " (saved in order shown in Technical 1 output)"
#> [193] " Number of Free Parameters"
#> [194] " H0 Loglikelihood"
#> [195] " H1 Loglikelihood"
#> [196] " Akaike (AIC)"
#> [197] " Bayesian (BIC)"
#> [198] " Sample-Size Adjusted BIC"
#> [199] " Chi-square : Value"
#> [200] " Chi-square : Degrees of Freedom"
#> [201] " Chi-square : P-Value"
#> [202] " CFI"
#> [203] " TLI"
#> [204] " RMSEA : Estimate"
#> [205] " RMSEA : Low CI"
#> [206] " RMSEA : High CI"
#> [207] " RMSEA : Probability"
#> [208] " SRMR"
#> [209] " Condition Number"
#> [210] ""
#> [211] " Save file"
#> [212] " file8ff6dd44a44"
#> [213] ""
#> [214] " Save file format Free"
#> [215] ""
#> [216] ""
#> [217] "SAVEDATA INFORMATION"
#> [218] ""
#> [219] ""
#> [220] " Estimated Covariance Matrix for the Parameter Estimates"
#> [221] ""
#> [222] " Save file"
#> [223] " file8ff10ac78d0"
#> [224] " Save format Free"
#> [225] ""
#> [226] " Beginning Time: 09:24:24"
#> [227] " Ending Time: 09:24:24"
#> [228] " Elapsed Time: 00:00:00"
#> [229] ""
#> [230] ""
#> [231] "Mplus VERSION 8.8 DEMO (Linux) has the following limitations:"
#> [232] " Maximum number of dependent variables: 6"
#> [233] " Maximum number of independent variables: 2"
#> [234] " Maximum number of between variables: 2"
#> [235] " Maximum number of continuous latent variables in time series analysis: 2"
#> [236] ""
#> [237] ""
#> [238] "MUTHEN & MUTHEN"
#> [239] "3463 Stoner Ave."
#> [240] "Los Angeles, CA 90066"
#> [241] ""
#> [242] "Tel: (310) 391-9971"
#> [243] "Fax: (310) 391-8971"
#> [244] "Web: www.StatModel.com"
#> [245] "Support: Support@StatModel.com"
#> [246] ""
#> [247] "Copyright (c) 1998-2022 Muthen & Muthen"Multiple Imputation
Parameters of the simple mediation model are estimated using normal theory maximum likelihood for each of the imputed data sets. The parameter estimates and their sampling covariance matrix are pooled.
fit_mi <- FitModelMI(
data_mi,
mplus_bin = mplus_bin
)
fit_mi
#> $coef
#> m~1 y~1 x~1 m~x y~m y~x
#> 0.005019703 0.332799419 0.412956299 0.377266653 0.150139986 0.153136568
#> m~~m y~~y x~~x
#> 1.211805873 1.146497984 0.242038727
#>
#> $vcov
#> m~1 y~1 x~1 m~x y~m
#> m~1 6.281569e-03 4.792023e-05 -9.324596e-05 -6.536580e-03 9.710301e-05
#> y~1 4.792023e-05 6.055600e-03 -6.372941e-05 -3.871823e-05 4.660348e-05
#> x~1 -9.324596e-05 -6.372941e-05 7.506654e-04 1.650903e-04 -8.964939e-06
#> m~x -6.536580e-03 -3.871823e-05 1.650903e-04 1.636016e-02 2.054703e-04
#> y~m 9.710301e-05 4.660348e-05 -8.964939e-06 2.054703e-04 3.194082e-03
#> y~x -6.691576e-05 -6.394353e-03 1.484601e-04 3.590605e-06 -1.068989e-03
#> m~~m 1.478045e-04 6.851728e-06 -3.582386e-05 -1.735251e-04 1.950598e-05
#> y~~y -3.229702e-05 5.311397e-05 -3.383887e-06 6.281239e-05 -1.973814e-05
#> x~~x 2.838276e-06 -1.069720e-05 1.271110e-05 5.582597e-06 2.221033e-05
#> y~x m~~m y~~y x~~x
#> m~1 -6.691576e-05 1.478045e-04 -3.229702e-05 2.838276e-06
#> y~1 -6.394353e-03 6.851728e-06 5.311397e-05 -1.069720e-05
#> x~1 1.484601e-04 -3.582386e-05 -3.383887e-06 1.271110e-05
#> m~x 3.590605e-06 -1.735251e-04 6.281239e-05 5.582597e-06
#> y~m -1.068989e-03 1.950598e-05 -1.973814e-05 2.221033e-05
#> y~x 1.623913e-02 9.434976e-05 -9.770793e-05 1.040343e-05
#> m~~m 9.434976e-05 8.866444e-03 2.301444e-05 -4.864670e-06
#> y~~y -9.770793e-05 2.301444e-05 7.891514e-03 -3.294018e-05
#> x~~x 1.040343e-05 -4.864670e-06 -3.294018e-05 3.692912e-04
#>
#> $vcov_tilde
#> m~1 y~1 x~1 m~x y~m
#> m~1 6.506328e-03 2.295136e-20 3.436532e-21 -6.513577e-03 -6.495798e-21
#> y~1 2.295136e-20 6.157602e-03 -3.839789e-21 -3.685758e-20 -1.408258e-05
#> x~1 3.436532e-21 -3.839789e-21 7.621849e-04 -1.200686e-20 -4.103301e-22
#> m~x -6.513577e-03 -3.685758e-20 -1.200686e-20 1.577610e-02 1.893660e-20
#> y~m -6.495798e-21 -1.408258e-05 -4.103301e-22 1.893660e-20 2.980379e-03
#> y~x -3.090516e-20 -6.160213e-03 6.016538e-21 6.635324e-20 -1.125836e-03
#> m~~m 1.131546e-06 -4.854797e-21 1.522422e-24 -2.193906e-06 2.148737e-20
#> y~~y -1.025535e-21 3.430788e-07 -3.969870e-25 2.327993e-21 4.900495e-08
#> x~~x 7.277823e-21 1.524870e-21 1.754247e-19 -1.740443e-20 6.227598e-22
#> y~x m~~m y~~y x~~x
#> m~1 -3.090516e-20 1.131546e-06 -1.025535e-21 7.277823e-21
#> y~1 -6.160213e-03 -4.854797e-21 3.430788e-07 1.524870e-21
#> x~1 6.016538e-21 1.522422e-24 -3.969870e-25 1.754247e-19
#> m~x 6.635324e-20 -2.193906e-06 2.327993e-21 -1.740443e-20
#> y~m -1.125836e-03 2.148737e-20 4.900495e-08 6.227598e-22
#> y~x 1.535902e-02 2.688057e-21 -6.866015e-07 -3.851020e-21
#> m~~m 2.688057e-21 9.252007e-03 -2.232232e-21 -1.277703e-22
#> y~~y -6.866015e-07 -2.232232e-21 8.281364e-03 1.063728e-23
#> x~~x -3.851020e-21 -1.277703e-22 1.063728e-23 3.691856e-04
#>
#> $vcov_between
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 4.405999e-04 4.744578e-05 -9.232273e-05 -6.866482e-04 9.614160e-05
#> [2,] 4.744578e-05 5.265992e-04 -6.309842e-05 -3.833489e-05 5.864989e-05
#> [3,] -9.232273e-05 -6.309842e-05 6.627736e-05 1.634557e-04 -8.876177e-06
#> [4,] -6.866482e-04 -3.833489e-05 1.634557e-04 2.186203e-03 2.034359e-04
#> [5,] 9.614160e-05 5.864989e-05 -8.876177e-06 2.034359e-04 5.153510e-04
#> [6,] -6.625323e-05 -8.596794e-04 1.469902e-04 3.555055e-06 -5.846278e-05
#> [7,] 1.453361e-04 6.783889e-06 -3.546917e-05 -1.698585e-04 1.931285e-05
#> [8,] -3.197724e-05 5.228337e-05 -3.350383e-06 6.219048e-05 -1.958624e-05
#> [9,] 2.810174e-06 -1.059128e-05 1.258525e-05 5.527323e-06 2.199042e-05
#> [,6] [,7] [,8] [,9]
#> [1,] -6.625323e-05 1.453361e-04 -3.197724e-05 2.810174e-06
#> [2,] -8.596794e-04 6.783889e-06 5.228337e-05 -1.059128e-05
#> [3,] 1.469902e-04 -3.546917e-05 -3.350383e-06 1.258525e-05
#> [4,] 3.555055e-06 -1.698585e-04 6.219048e-05 5.527323e-06
#> [5,] -5.846278e-05 1.931285e-05 -1.958624e-05 2.199042e-05
#> [6,] 2.436800e-03 9.341561e-05 -9.613070e-05 1.030043e-05
#> [7,] 9.341561e-05 5.612308e-04 2.278657e-05 -4.816505e-06
#> [8,] -9.613070e-05 2.278657e-05 4.580570e-04 -3.261404e-05
#> [9,] 1.030043e-05 -4.816505e-06 -3.261404e-05 3.773239e-05
#>
#> $vcov_within
#> m~1 y~1 x~1 m~x y~m
#> m~1 5.836563e-03 2.058873e-20 3.082773e-21 -5.843066e-03 -5.827117e-21
#> y~1 2.058873e-20 5.523735e-03 -3.444519e-21 -3.306344e-20 -1.263291e-05
#> x~1 3.082773e-21 -3.444519e-21 6.837252e-04 -1.077087e-20 -3.680905e-22
#> m~x -5.843066e-03 -3.306344e-20 -1.077087e-20 1.415210e-02 1.698725e-20
#> y~m -5.827117e-21 -1.263291e-05 -3.680905e-22 1.698725e-20 2.673577e-03
#> y~x -2.772376e-20 -5.526077e-03 5.397193e-21 5.952281e-20 -1.009942e-03
#> m~~m 1.015064e-06 -4.355042e-21 1.365703e-24 -1.968064e-06 1.927545e-20
#> y~~y -9.199656e-22 3.077621e-07 -3.561210e-25 2.088349e-21 4.396036e-08
#> x~~x 6.528640e-21 1.367899e-21 1.573664e-19 -1.561281e-20 5.586526e-22
#> y~x m~~m y~~y x~~x
#> m~1 -2.772376e-20 1.015064e-06 -9.199656e-22 6.528640e-21
#> y~1 -5.526077e-03 -4.355042e-21 3.077621e-07 1.367899e-21
#> x~1 5.397193e-21 1.365703e-24 -3.561210e-25 1.573664e-19
#> m~x 5.952281e-20 -1.968064e-06 2.088349e-21 -1.561281e-20
#> y~m -1.009942e-03 1.927545e-20 4.396036e-08 5.586526e-22
#> y~x 1.377796e-02 2.411347e-21 -6.159224e-07 -3.454594e-21
#> m~~m 2.411347e-21 8.299601e-03 -2.002445e-21 -1.146176e-22
#> y~~y -6.159224e-07 -2.002445e-21 7.428876e-03 9.542269e-24
#> x~~x -3.454594e-21 -1.146176e-22 9.542269e-24 3.311815e-04
#>
#> $ariv
#> [1] 0.1147533
#>
#> $m
#> [1] 100
#>
#> $k
#> [1] 9
#>
#> $nu1
#> [1] 9
#>
#> $nu2
#> [1] 83372.16
#>
#> $d1
#> [1] 86.11739Monte Carlo Method
Maximum Likelihood
mc_ml <- MCML(
fit_ml,
R = R
)
mc_ml
#> 0.05% 0.5% 2.5% 97.5% 99.5%
#> -0.0140360145 0.0001377727 0.0094455357 0.1233982450 0.1509750184
#> 99.95%
#> 0.1878755830Multiple Imputation
The two versions are based on the pooled covariance matrix used.
vcov uses the total covariance matrix \(\mathbf{V}_{T}\) while
vcov_tilde uses the adjusted total covariance matrix \(\tilde{\mathbf{V}}_{T}\).
mc_mi <- MCMI(
fit_mi,
R = R
)
mc_mi
#> 0.05% 0.5% 2.5% 97.5% 99.5% 99.95%
#> vcov -0.014975761 -0.001060201 0.008691238 0.1244539 0.1533265 0.2002091
#> vcov_tilde -0.009596199 0.001044739 0.010304272 0.1212996 0.1455524 0.1753921Nonparametric Bootstrapping
Maximum Likelihood Nested within Nonparametric Bootstrap (NB(ML))
nb_ml <- NBML(
data_missing,
B = B,
mplus_bin = mplus_bin
)
nb_ml
#> Lower .5% Lower 2.5% Lower 5% Estimate Upper 5% Upper 2.5% Upper .5%
#> bc 0.002 0.012 0.018 0.056 0.127 0.142 0.171
#> pc -0.001 0.008 0.013 0.056 0.114 0.129 0.160Multiple Imputation
Multiple Imputation Nested Within Nonparametric Bootstrap (NB(MI))
nb_mi <- NBMI(
data_missing,
data_mi,
B = B,
m = m,
mplus_bin = mplus_bin
)
nb_mi
#> 0.05% 0.5% 2.5% 97.5% 99.5% 99.95%
#> bc -0.006549636 0.002779928 0.012916304 0.1429212 0.1703444 0.2140131
#> pc -0.010102987 -0.001063635 0.007508727 0.1272201 0.1598429 0.1940216Nonparametric Bootstrap Nested Within Multiple Imputation (MI(NB))
mi_nb <- MINB(
data_mi,
B = B
)
mi_nb
#> 0.05% 0.5% 2.5% 97.5% 99.5% 99.95%
#> bc -0.009698788 0.0026598776 0.0123565 0.1410420 0.1716812 0.2120889
#> pc -0.015155326 -0.0006408286 0.0085043 0.1298082 0.1603919 0.1994659Joint Significance Test
The JointSigML() and JointSigMI() functions
return 1 for a significant result and 0
otherwise. Note that the output vector’s name corresponds to the
alpha level. The default value of alpha is
alpha = c(0.05, 0.01, 0.001).
Maximum Likelihood
joint_ml <- JointSigML(
fit_ml
)
joint_ml
#> 0.05 0.01 0.001
#> 1 1 0Multiple Imputation
The two versions are based on the two types of pooled covariance matrices \(\mathbf{V}_{T}\) and \(\tilde{\mathbf{V}}_{T}\).
joint_mi <- JointSigMI(
fit_mi
)
joint_mi
#> 0.05 0.01 0.001
#> vcov 1 1 0
#> vcov_tilde 1 1 0Results
Here is a summary of the \(95\%\) confidence intervals.
ci <- rbind(
mc_ml[c(3, 4)],
mc_mi[, c(3, 4)],
nb_ml[, c(2, 6)],
nb_mi[, c(3, 4)],
mi_nb[, c(3, 4)]
)
rownames(ci) <- c(
"MC.FIML",
"MC.MI",
"MC.MI.ADJ",
"NBBC(FIML)",
"NBPC(FIML)",
"NBBC(MI)",
"NBPC(MI)",
"MI(NBBC)",
"MI(NBPC)"
)
knitr::kable(ci)| 2.5% | 97.5% | |
|---|---|---|
| MC.FIML | 0.0094455 | 0.1233982 |
| MC.MI | 0.0086912 | 0.1244539 |
| MC.MI.ADJ | 0.0103043 | 0.1212996 |
| NBBC(FIML) | 0.0120000 | 0.1420000 |
| NBPC(FIML) | 0.0080000 | 0.1290000 |
| NBBC(MI) | 0.0129163 | 0.1429212 |
| NBPC(MI) | 0.0075087 | 0.1272201 |
| MI(NBBC) | 0.0123565 | 0.1410420 |
| MI(NBPC) | 0.0085043 | 0.1298082 |
References
Elliot, D. L., Goldberg, L., Kuehl, K. S., Moe, E. L., Breger, R. K., & Pickering, M. A. (2007). The PHLAME (Promoting Healthy Lifestyles: Alternative Models’ Effects) firefighter study: Outcomes of two models of behavior change. Journal of occupational and environmental medicine, 49(2), 204–213. http://doi.org/10.1097/JOM.0b013e3180329a8d
Wu, W., & Jia, F. (2013). A new procedure to test mediation with missing data through nonparametric bootstrapping and multiple imputation, Multivariate Behavioral Research, 48(5), 663-691. http://doi.org/10.1080/00273171.2013.816235
Yuan, Y., & MacKinnon, D. P. (2009). Bayesian mediation analysis. Psychological methods, 14(4), 301–322. http://doi.org/10.1037/a0016972