[R-sig-ME] weighted vs unweighted GLMER variance estimates

Mon Jan 7 18:38:01 CET 2013

Dear Users,

I've recently been working on analyzing a longitudinal & cross-sectional fMRI data set of unbalanced design consisting of approx 160 participants. Consistent with multilevel modeling often used in fMRI analysis, I've been using parameter and variance estimates in neural response from each subject as my data, with Age and Age^2 as predictor variables. I've also been checking to see how the results from weighted GLMs (wGLM), weighted GLMER (wGLMER), and unweighted GLMs, compare to a standard fMRI analysis package (FSL) with a subset of the data. In doing so, I've found that the weighted GLMER produces more similar variance estimates than the unweighted GLMER, provided I make an adjustment to output provided by lme4.

Essentially, the fixed effects error variance for the wGLMER appears to be larger than the error variance for the GLMER on an order proportional to error of the between-subjects residuals. This adjustment appears to be reliable across the 1900 regressions I ran my tests on.

I'm unsure of why using weights should so drastically affect the Fixed Effects variance, why dividing the wGLMER variance estimates by the residual variance achieves similarity to the unweighted GLMER variance estimates, and if I am justified in using these adjusted amounts to calculate t-values. I am including data from my full sample for an example, but am printing only some output here for immediate observation.

Help on this matter would be very much appreciated.

Regards,
David Paulsen

> # DISPLAY RESULTS
> summary(glmer_out)
Linear mixed model fit by REML
Formula: yhat ~ Age_adj + I(Age_adj^2) + (1 | subjectID)
  AIC  BIC logLik deviance REMLdev
 2714 2731  -1352     2710    2704
Random effects:
 Groups    Name        Variance Std.Dev.
 subjectID (Intercept)  325.64  18.045
 Residual              2846.26  53.350
Number of obs: 249, groups: subjectID, 137

Fixed effects:
             Estimate Std. Error t value
(Intercept)   31.8412     4.7634   6.685
Age_adj        2.7313     1.4915   1.831
I(Age_adj^2)  -0.4337     0.3100  -1.399

Correlation of Fixed Effects:
            (Intr) Age_dj
Age_adj      0.039
I(Age_dj^2) -0.507 -0.615
> summary(wglmer_out)
Linear mixed model fit by REML
Formula: yhat ~ Age_adj + I(Age_adj^2) + (1 | subjectID)
   AIC   BIC logLik deviance REMLdev
 751.9 769.4 -370.9    747.5   741.9
Random effects:
 Groups    Name        Variance Std.Dev.
 subjectID (Intercept) 818725.8 904.835
 Residual                2400.1  48.991
Number of obs: 249, groups: subjectID, 137

Fixed effects:
             Estimate Std. Error t value
(Intercept)   30.1223   215.3330   0.140
Age_adj        2.8065    66.2379   0.042
I(Age_adj^2)  -0.4986    13.2487  -0.038

Correlation of Fixed Effects:
            (Intr) Age_dj
Age_adj      0.056
I(Age_dj^2) -0.506 -0.626
>
> # COMPARE GLMER SE, wGLMER SE, & wGLMER ADJUSTED SE
> glmer_stderr
 (Intercept)      Age_adj I(Age_adj^2)
   4.7634370    1.4914602    0.3099561
> wglmer_stderr
 (Intercept)      Age_adj I(Age_adj^2)
   215.33304     66.23788     13.24866
> wglmer_stderr_adj
 (Intercept)      Age_adj I(Age_adj^2)
   4.3953591    1.3520419    0.2704305

# DATA & CODE
yhat <- c(-17.86383, -67.93507, 62.77558, -25.00129, 8.600049, 19.74757, 36.35019, 19.78256, 63.44397, 187.6141, 39.21401, -32.13604, 112.21, 99.78252, 52.1128, 8.983917, -0.2228356, 45.04539, 152.8446, 69.4516, -17.09545, 33.01205, 11.79658, 236.2675, 59.41111, 48.96542, -28.8011, 8.018624, -80.94499, 61.5362, 122.6593, 26.82249, 137.6959, 13.70128, 153.4673, 88.05206, 12.78811, 9.1956, 19.80628, 38.78817, 54.4374, -26.908, -17.47678, -12.40804, -33.34377, -12.54371, -29.12496, 29.89239, 76.09109, -1.456215, -42.91579, -35.48977, 40.3413, 96.75193, -26.20043, 72.88014, -4.498486, 44.04082, 120.6856, 36.17123, -36.29802, 53.01909, -51.50906, -22.85605, 49.57092, 120.6811, 114.3392, -2.821136, 81.09837, -55.07222, 73.61099, 47.53521, 49.42628, 145.0036, 40.72733, 2.6586, -23.1839, 74.15407, 34.09974, 43.26672, 34.73029, -43.57184, -19.39236, 99.23603, -8.092349, -9.658018, 46.68232, 18.92779, 60.55532, 67.58183, 70.43863, -66.24371, 30.25297, 63.24903, -44.24773, 110.2207, 51.43445, 34.92665, -9.43949, 33.45405, 31.89272, 14.7246, -37.78331, 101.5455, -6.926058, 15.54408, 200.0708, 49.00461, 8.489271, 15.56577, 18.82096, 79.90823, -33.52191, 65.90131, -41.56767, 41.44206, 17.85847, -0.7615868, -17.49322, 21.56572, -55.35418, -96.07809, -33.83941, 20.94368, 94.30896, 82.08093, -30.9233, 28.62884, -9.951429, 80.26689, 101.3565, 57.31608, 33.58985, 6.77054, 67.09788, -8.760614, -6.980731, -5.862709, 54.40738, 43.92796, 39.94324, 4.664509, 27.7117, 9.358894, 25.28339, -15.61768, 86.86378, 83.1256, -21.28656, -72.29863, 9.106373, 35.14648, 25.62542, 91.91111, 51.71715, 62.60024, 261.3518, 51.44857, -23.74933, 81.17366, -0.2988409, 25.64609, 0.03400882, -33.95691, 37.02612, 50.46914, -24.85151, 27.13361, 48.47046, 40.92603, 68.63319, 49.97451, -2.729544, -76.28941, 64.21523, -39.13202, 106.9234, 69.32959, 55.89867, 85.48119, 51.0333, -12.99842, 46.53439, 49.33897, -3.042722, 110.3458, 65.66805, 119.5013, -56.69803, 36.52827, -20.95306, 37.66682, -7.274448, 90.77976, 55.46455, -50.28808, 59.24579, 50.87093, -34.55978, 46.75426, -4.968956, -58.17086, 21.63069, 23.14527, -8.624804, 130.3394, 25.99968, 26.47401, 9.909436, -42.20339, 33.77307, -42.8187, -33.09747, -4.354581, 19.64606, 49.20227, 27.82024, 25.82506, 110.3884, 110.5982, 185.3917, -75.21708, 40.95378, 51.64703, -67.93621, 95.71213, 32.24789, 3.461251, -37.93186, -25.56975, 41.39754, 69.54233, 50.13354, -39.70442, 56.94862, -27.39709, 96.96685, 74.49326, 16.48888, -13.52116, 3.87862, 105.7628, -70.72237, 35.75502, 77.68787, 102.3832, -40.57845, 104.7736, 12.21588)

Age_adj <- c(5.78, 10.55, 0.41, 2.01, -1.67, -0.01, 0.69, 2.16, 1.09, 2.58, 4.02, 0.56, 2.03, -0.25, 1.16, 5.11, 5.45, 0.59, 2.12, 3.86, 8.65, 1.6, -4.51, -2.83, 1.32, 5.07, 4.21, 3.72, 5.29, 3.48, 4.91, 6.47, 3.93, 5.79, 7.12, 8.62, 1.86, 3.43, 5.78, 3.09, 4.66, 6.15, 8.55, -3.59, -2.09, 2.03, 3.52, 4.38, 7.91, 2.53, 4.22, 5.66, -0.18, 1.41, 3.01, 3.8, 6.79, 6.79, 3.96, -2.35, -0.76, 1.3, -0.15, 1.62, 5.5, 0.65, 3.85, -1.13, 0.35, -3.29, -1.81, 0.39, -3.94, -2.53, -1.01, -4.43, -2.79, -1.28, -4.89, -2.12, -0.85, 0.65, 2.37, 3.98, -0.43, 1.45, 2.49, -3.47, -1.92, -0.55, 2.91, -1.47, 1.68, -1.03, 0.73, 4.18, 5.58, -2.53, -0.87, 2.46, 5.72, -0.76, 3.31, -0.16, 1.83, 3, 1.16, 2.69, 0.69, 2.2, 3.68, 1.1, 3.17, 4.92, 6.45, 0.56, 2.13, 3.79, 5.25, 0.41, 1.96, 3.51, -0.11, 1.44, 2.9, -3.76, -2.12, -0.66, 1.68, 3.2, 3.6, 3.14, 5.17, 2.86, 4.89, 2.44, -0.26, 1.52, -4.28, -2.73, -1.18, -0.39, 0.9, -3.88, -2.09, 2.88, 2.96, 4.58, 5.9, -2.88, -1.35, 0.12, 3.39, 4.88, 6.25, 3.57, 5.33, 3.49, 5.31, 6.43, 3.9, 5.69, 3.33, -3.97, -1.62, -0.78, -1.32, 0.2, -1.56, -3.23, -2.59, -1.19, 3.86, 3.88, 5.59, -3.31, 3.74, 5.47, -3.85, 0.88, 2.64, 1.28, -2.77, -1.19, -3.24, 3.54, 3.81, -1.35, 0.17, 3.86, 5.42, -4.54, -2.85, -3.72, 1.69, -2.22, -3.3, -1.7, -1.06, 0.52, -0.88, -2.84, -1.13, 2.65, -4.12, -2.07, 1.2, -4.84, -3.1, -4.87, -3.18, -1.41, -1.95, 0.18, -1.62, 1.67, 3.15, 0.07, 1.53, 3.65, 5.48, 0.04, -4.08, -2.44, 0.74, -0.23, 1.44, 1.94, 3.68, -1.19, 3.11, 4.9, 3.95, 5.73, 0.6, 2.31, 3.9, 3.86, 5.57, -0.78, 0.71, 1.21, 3.32, 0.97, 1.29, 2.91, 0.3, 1.74, -1.86)

subjectID <- c(10128, 10134, 10152, 10152, 10153, 10153, 10156, 10156, 10173, 10173, 10173, 10181, 10181, 10300, 10300, 10315, 10385, 10406, 10406, 10406, 10431, 10451, 10466, 10466, 10529, 10558, 10565, 10567, 10567, 10568, 10568, 10568, 10572, 10572, 10572, 10574, 10585, 10585, 10585, 10589, 10589, 10589, 10594, 10599, 10599, 10604, 10604, 10604, 10605, 10608, 10608, 10608, 10616, 10616, 10616, 10623, 10623, 10623, 10625, 10626, 10626, 10626, 10627, 10627, 10629, 10633, 10635, 10636, 10636, 10637, 10637, 10637, 10638, 10638, 10638, 10644, 10644, 10644, 10646, 10652, 10653, 10653, 10654, 10654, 10661, 10661, 10661, 10662, 10662, 10662, 10664, 10665, 10665, 10666, 10666, 10667, 10667, 10671, 10671, 10673, 10673, 10674, 10675, 10677, 10677, 10677, 10678, 10678, 10680, 10680, 10680, 10686, 10689, 10689, 10689, 10696, 10697, 10697, 10697, 10699, 10699, 10699, 10700, 10700, 10700, 10701, 10701, 10701, 10703, 10703, 10707, 10708, 10708, 10709, 10709, 10710, 10711, 10711, 10717, 10717, 10717, 10747, 10747, 10749, 10749, 10757, 10758, 10758, 10758, 10760, 10760, 10760, 10761, 10761, 10761, 10762, 10762, 10765, 10765, 10765, 10766, 10766, 10768, 10772, 10772, 10773, 10774, 10774, 10778, 10779, 10780, 10780, 10782, 10786, 10786, 10787, 10788, 10788, 10790, 10796, 10796, 10797, 10798, 10798, 10800, 10802, 10803, 10804, 10804, 10805, 10805, 10806, 10806, 10807, 10808, 10809, 10811, 10811, 10812, 10812, 10813, 10814, 10814, 10817, 10818, 10818, 10820, 10821, 10821, 10822, 10822, 10824, 10826, 10826, 10836, 10839, 10839, 10840, 10840, 10841, 10841, 10842, 10843, 10843, 10845, 10847, 10847, 10849, 10849, 10850, 10851, 10851, 10852, 10852, 10869, 10869, 10870, 10872, 10872, 10873, 10873, 10874, 10875, 10877, 10879, 10879, 10887, 10887, 10888)

var_est <- c(2117.232, 1856.527, 2752.249, 1364.204, 1195.229, 4033.198, 6462.032, 1612.014, 2854.178, 3227.569, 2090.262, 2407.377, 2311.557, 3347.094, 2348.126, 1691.455, 3862.144, 1361.611, 5045.376, 2004.46, 2111.046, 1531.715, 4047.582, 4162.438, 1782.844, 2445.805, 4974.277, 972.256, 2732.625, 3149.915, 4282.503, 1667, 3445.044, 3929.609, 1857.306, 1929.016, 1806.192, 1350.487, 2163.956, 1382.237, 2105.488, 2425.088, 1585.392, 1284.99, 1839.946, 1783.923, 3088.686, 1905.855, 1395.401, 3912.845, 1873.235, 2499.939, 1997.997, 3953.164, 1110.427, 2128.916, 1421.97, 2578.543, 2438.239, 2143.414, 5448.135, 895.8371, 3876.441, 2454.641, 1266.05, 1286.321, 2371.491, 1845.233, 948.009, 2334.511, 1238.893, 2484.186, 2104.395, 1628.669, 3306.381, 1225.193, 2251.642, 2629.077, 2847.037, 3861.317, 2171.143, 1803.323, 4486.976, 3057.108, 1319.224, 2902.172, 2046.981, 2164.243, 3769.546, 1308.071, 2630.981, 1733.245, 1859.789, 2422.816, 5358.247, 5088.674, 1326.271, 2338.529, 1818.681, 1583.202, 1830.168, 2420.32, 1737.724, 2471.06, 1173.188, 2552.61, 3719.981, 1856.206, 1467.039, 4748.495, 2014.772, 1903.904, 1797.682, 2616.952, 2618.1, 2875.845, 1561.778, 2121.504, 3907.931, 1736.165, 5158.229, 2934.292, 2275.788, 2782.14, 7164.727, 3635.709, 3204.541, 3073.852, 1856.379, 2473.882, 4525.398, 2001.25, 2475.327, 2637.609, 7965.129, 5800.15, 3788.97, 3192.963, 1291.541, 6677.519, 2196.154, 2828.934, 2965.4, 3243.802, 3391.377, 2682.823, 1832.081, 2508.383, 3403.081, 2939.152, 3820.748, 2516.928, 2568.668, 5601.77, 2319.515, 3408.002, 7352.196, 3070.829, 1270.897, 2475.052, 2129.083, 3462.514, 2968.115, 1359.369, 2687.299, 2150.726, 1933.411, 1238.447, 4716.554, 3671.501, 4155.547, 2423.732, 1733.876, 2068.158, 2195.333, 1946.794, 3901.803, 1843.907, 2325.319, 3433.175, 2690.929, 2712.109, 3240.584, 2034.036, 814.4943, 4247.212, 4431.068, 1880.548, 1985.781, 2672.374, 2516.915, 8474.012, 2994.591, 3063.671, 3094.606, 1464.994, 1020.753, 3616.184, 1180.809, 3700.844, 2663.671, 2229.899, 1616.548, 2157.592, 2610.113, 5337.962, 1900.459, 1621.1, 2519.966, 1817.123, 2161.079, 5065.147, 1928.294, 2759.455, 2299.465, 2884.927, 3980.114, 5071.609, 2488.129, 2993.238, 1656.281, 2927.721, 2920.717, 1268.808, 3921.836, 2113.641, 4456.005, 3496.747, 3684.217, 2233.274, 2769.909, 1703.298, 11747.37, 1467.061, 3739.88, 4333.55, 4014.81, 2267.72, 1952.689, 2909.576, 4136.469, 4080.664, 3515.684, 1847.262, 2734.856, 4158.62, 6032.312, 2431.275, 2705.765)

# RUN GLMER, SAVING STD.ERR.
glmer_out <- glmer(yhat ~ Age_adj + I(Age_adj^2) + (1|subjectID))
glmer_stderr <- coef(summary(glmer_out))[,"Std. Error"]

# RUN wGLMER, SAVING STD.ERR.
wglmer_out <- glmer(yhat ~ Age_adj + I(Age_adj^2) + (1|subjectID), weights=1/var_est)
rand_resid_sd <- as.numeric(summary(wglmer_out)@REmat[2,"Std.Dev."])

wglmer_stderr <- coef(summary(wglmer_out))[,"Std. Error"]
wglmer_stderr_adj <- wglmer_stderr/rand_resid_sd

# DISPLAY RESULTS
summary(glmer_out)
summary(wglmer_out)

# COMPARE GLMER SE, wGLMER SE, & wGLMER ADJUSTED SE
glmer_stderr
wglmer_stderr
wglmer_stderr_adj

David J. Paulsen, Ph.D.
Laboratory of Neurocognitive Development
Western Psychiatric Institute and Clinic
University of Pittsburgh Medical Center
Loeffler Building
121 Meyran Avenue
Pittsburgh, PA 15213
412.383.8168