Re: [R] lmer and mixed effects logistic regression

From: Rick Bilonick <rab45+_at_pitt.edu>
Date: Thu 22 Jun 2006 - 09:01:50 EST

On Wed, 2006-06-21 at 08:35 -0700, Spencer Graves wrote:
> You could think of 'lmer(..., family=binomial)' as doing a separate
> "glm" fit for each subject, with some shrinkage provided by the assumed
> distribution of the random effect parameters for each subject. Since
> your data are constant within subject, the intercept in your model
> without the subject's random effect distribution will be estimated at
> +/-Inf. Since this occurs for all subjects, the maximum likelihood
> estimate of the subject variance is Inf, which is what I wrote in an
> earlier contribution to this thread.
>
> What kind of answer do you get from SAS NLMIXED? If it does NOT tell
> you that there is something strange about the estimation problem you've
> given it, I would call that a serious infelicity in the code. If it is
> documented behavior, some might argue that it doesn't deserve the "B"
> word ("Bug"). The warning messages issued by 'lmer' in this case are
> something I think users would want, even if they are cryptic.
>
> Hope this helps.
> Spencer Graves
>
I did send in an example with data set that duplicates the problem. Changing the control parameters allowed lmer to produce what seem like reasonable estimates. Even for the case with essentially duplicate pairs, lmer and NLMIXED produce similar estimates (finite intercepts also) although lmer's coefficient estimates are as far as I can tell the same as glm but the standard errors are larger.

The problem I really want estimates for is different from this one explanatory factor example. The model I estimate will have several explanatory factors, including factors that differ within each subject (although the responses within each subject are the same). BTW, as far as I know, the responses could be different within a subject but it seems to be very rare.

Possibly the example I thought I sent never made it to the list. The example is below.

Rick B.

###########################################################################
# Example of lmer error message

I made an example data set that exhibits the error. There is a dump of the data frame at the end.

First, I updated all my packages:

> sessionInfo()

Version 2.3.1 (2006-06-01)
i686-redhat-linux-gnu

attached base packages:
[1] "methods" "stats" "graphics" "grDevices" "utils" "datasets"
[7] "base"

other attached packages:

     chron lme4 Matrix lattice    "2.3-3" "0.995-2" "0.995-11" "0.13-8"

But I still get the error.

For comparison, here is what glm gives:

> summary(glm(y~x,data=example.df,family=binomial))

Call:
glm(formula = y ~ x, family = binomial, data = example.df)

Deviance Residuals:

    Min 1Q Median 3Q Max -1.6747 -0.9087 -0.6125 1.1447 2.0017

Coefficients:

            Estimate Std. Error z value Pr(>|z|)
(Intercept)  -0.4786     0.1227  -3.901 9.59e-05 ***
x             0.7951     0.1311   6.067 1.31e-09 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 436.63  on 324  degrees of freedom
Residual deviance: 394.15  on 323  degrees of freedom
AIC: 398.15

Number of Fisher Scoring iterations: 4


Running lmer without any tweaks:


> (lmer(y~(1|id)+x,data=example.df,family=binomial))
Error in lmer(y ~ (1 | id) + x, data = example.df, family = binomial) : Leading minor of order 2 in downdated X'X is not positive definite In addition: Warning message: nlminb returned message singular convergence (7) in: LMEopt(x = mer, value = cv) Running lmer with list(msVerbose=TRUE):
> (lmer(y~(1|
id)+x,data=example.df,family=binomial,control=list(msVerbose=TRUE))) 0 -545.002: 44801.6 1 -545.002: 44801.6 2 -545.002: 44801.6 3 -545.003: 44801.9 4 -545.014: 44805.2 5 -545.123: 44838.3 6 -546.208: 45168.3 7 -556.572: 48444.8 8 -628.932: 78993.4 9 -699.716: 127441. 10 -771.102: 206437. 11 -842.258: 333880. 12 -913.501: 540319. 13 -984.712: 874202. 14 -1055.93: 1.41452e+06 15 -1127.15: 2.28873e+06 16 -1198.37: 3.70326e+06 17 -1269.59: 5.99199e+06 18 -1340.81: 9.69524e+06 19 -1412.03: 1.56872e+07 20 -1483.25: 2.53825e+07 21 -1554.47: 4.10697e+07 22 -1625.69: 6.64522e+07 23 -1696.91: 1.07522e+08 24 -1768.13: 1.73974e+08 25 -1839.35: 2.81496e+08 26 -1910.57: 4.55470e+08 27 -1981.78: 7.36966e+08 28 -2053.00: 1.19244e+09 29 -2124.22: 1.92940e+09 30 -2195.44: 3.12184e+09 31 -2266.66: 5.05124e+09 32 -2337.88: 8.17308e+09 33 -2409.10: 1.32243e+10 34 -2480.32: 2.13974e+10 35 -2551.54: 3.46217e+10 36 -2622.76: 5.60190e+10 37 -2693.98: 9.06405e+10 38 -2765.20: 1.46659e+11 39 -2836.42: 2.37299e+11 40 -2907.64: 3.83962e+11 41 -2978.85: 6.21253e+11 42 -3050.07: 1.00521e+12 43 -3121.28: 1.62645e+12 44 -3192.47: 2.63147e+12 45 -3263.70: 4.25757e+12 46 -3334.89: 6.88953e+12 47 -3406.11: 1.11441e+13 48 -3477.22: 1.80392e+13 49 -3548.36: 2.91492e+13 50 -3619.76: 4.72269e+13 51 -3690.52: 7.63668e+13 52 -3761.36: 1.23295e+14 53 -3832.63: 1.99577e+14 54 -3900.88: 3.22856e+14 55 -3968.08: 4.97009e+14 0 -4067.06: 1.67844e+15 1 -4067.06: 1.67844e+15 0 -4265.60: 5.77607e+15 1 -4265.60: 5.77607e+15 0 -4474.52: 1.96098e+16 1 -4474.52: 1.96098e+16 0 -4723.57: 6.68597e+16 1 -4723.57: 6.68597e+16 0 -4985.37: 2.20089e+17 1 -4985.37: 2.20089e+17 0 -5268.68: 7.69417e+17 1 -5268.68: 7.69417e+17 0 -5536.64: 2.48775e+18 1 -5536.64: 2.48775e+18 0 -5853.10: 8.45248e+18 1 -5853.10: 8.45248e+18 0 -6197.46: 3.00106e+19 1 -6197.46: 3.00106e+19 0 -6400.09: 8.72855e+19 1 -6400.09: 8.72855e+19 0 -6769.87: 3.19354e+20 1 -6769.87: 3.19354e+20 0 -7085.60: 1.14993e+21 1 -7085.60: 1.14993e+21 0 -7414.58: 4.43964e+21 1 -7414.58: 4.43964e+21 0 -7665.61: 1.61085e+22 1 -7665.61: 1.61085e+22 Error in lmer(y ~ (1 | id) + x, data = example.df, family = binomial, : Leading minor of order 2 in downdated X'X is not positive definite In addition: Warning message: nlminb returned message singular convergence (7) in: LMEopt(x = mer, value = cv) Running lmer with method="Laplace" and control=list(usePQL=FALSE,msVerbose=TRUE):
> (lmer(y~(1|id)+x,data=example.df,family=binomial,method="Laplace",
+ control=list(usePQL=FALSE,msVerbose=TRUE))) 0 347.321: -0.478643 0.795145 1.45231 1 334.637: -0.775380 1.49795 2.09885 2 326.045: -0.631955 0.917513 2.90042 3 307.930: -0.627581 1.85085 4.66928 4 304.717: -1.06671 1.40101 5.11069 5 299.588: -1.05336 1.85102 5.73305 6 297.157: -0.682292 1.60623 6.35949 7 282.629: -1.33421 1.86152 10.2167 8 270.279: -1.44945 2.72297 14.8450 9 263.248: -1.61188 3.21518 19.5257 10 254.336: -1.89092 4.01520 29.0932 11 248.253: -2.13096 4.72573 39.9024 12 243.359: -2.39747 5.49392 53.8331 13 239.255: -2.66754 6.31763 71.9027 14 235.865: -2.91894 7.17523 94.3541 15 232.831: -3.14279 8.11396 123.501 16 230.229: -3.32800 9.12440 159.978 17 227.957: -3.45824 10.1876 205.312 18 225.987: -3.50977 11.2006 258.137 19 223.822: -3.42383 12.2016 327.929 20 222.281: -3.29714 12.9668 393.939 21 218.687: -2.35417 15.1107 657.987 22 217.978: -2.00284 15.3087 724.381 23 216.828: -1.03243 15.3436 883.159 24 216.641: -0.727910 15.0860 924.584 25 216.561: -0.634457 14.8052 935.901 26 216.477: -0.670831 14.4966 934.259 27 216.335: -0.882568 14.1066 925.552 28 216.153: -1.24388 13.9061 926.647 29 215.914: -1.70066 14.0769 966.092 30 215.643: -2.07605 14.7379 1073.14 31 215.365: -2.25220 15.8379 1261.63 32 215.169: -2.20650 16.9633 1485.79 33 215.065: -2.05998 17.7714 1685.40 34 214.993: -1.85386 18.2239 1859.43 35 214.948: -1.69235 18.3198 1985.48 36 214.933: -1.65586 18.2629 2051.34 37 214.933: -1.65578 18.2629 2051.34 38 214.933: -1.65579 18.2629 2051.34 39 214.933: -1.65586 18.2629 2051.34 40 214.933: -1.65654 18.2625 2051.34 41 214.932: -1.66423 18.2585 2051.33 42 214.931: -1.70783 18.2351 2051.33 43 214.931: -1.73215 18.2201 2051.43 44 214.931: -1.74205 18.2078 2051.65 45 214.930: -1.73708 18.2686 2076.43 46 214.929: -1.73209 18.3805 2120.39 47 214.929: -1.73283 18.3612 2112.76 48 214.929: -1.73334 18.3600 2112.79 49 214.929: -1.73332 18.3600 2112.79 50 214.929: -1.73332 18.3600 2112.79 51 214.929: -1.73332 18.3600 2112.79 52 214.929: -1.73332 18.3600 2112.79 53 214.929: -1.73332 18.3600 2112.79 54 214.929: -1.73332 18.3600 2112.79 Generalized linear mixed model fit using Laplace Formula: y ~ (1 | id) + x Data: example.df Family: binomial(logit link) AIC BIC logLik deviance 220.9293 232.2807 -107.4646 214.9293 Random effects: Groups Name Variance Std.Dev. id (Intercept) 2112.8 45.965 number of obs: 325, groups: id, 177 Estimated scale (compare to 1) 0.06664838 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -1.7333 5.7142 -0.30333 0.76164 x 18.3600 7.3318 2.50416 0.01227 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Correlation of Fixed Effects: (Intr) x -0.382 Note that the results for x don't agree at all with what glm outputs. The estimated scale is very small and the sd for id appears to be very large. Now changing method="Laplace" to method="ML":
> (lmer(y~(1|id)+x,data=example.df,family=binomial,method="ML",
+ control=list(usePQL=FALSE,msVerbose=TRUE))) Generalized linear mixed model fit using PQL Formula: y ~ (1 | id) + x Data: example.df Family: binomial(logit link) AIC BIC logLik deviance 353.3209 364.6724 -173.6604 347.3209 Random effects: Groups Name Variance Std.Dev. id (Intercept) 1.4523 1.2051 number of obs: 325, groups: id, 177 Estimated scale (compare to 1) 0.2372670 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -0.47864 0.16114 -2.9703 0.002975 ** x 0.79514 0.16872 4.7128 2.444e-06 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Correlation of Fixed Effects: (Intr) x -0.129 The estimated coefficients are the same as glm to 4 decimal places. The se's are about 30% larger than for glm. The sd for id is much smaller and the scale is larger. If I try to turn PQL back on I get the error message. I used ML and PQL off on the original data set and the results are ROUGHLY similar to what SAS NLMIXED gives but the coefficient for x is about 20% lower than NLMIXED. I haven't had a chance to run NLMIXED on the example data frame yet. Finally, besides my thanks for the help and apologies for the length of this post, here is the dump of the data frame: example.df <- structure(list(id = structure(as.integer(c(1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 55, 56, 56, 57, 57, 58, 58, 59, 59, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 67, 67, 68, 69, 69, 70, 70, 71, 71, 72, 72, 73, 73, 74, 75, 75, 76, 76, 77, 77, 78, 78, 79, 79, 80, 81, 81, 82, 82, 83, 83, 84, 85, 85, 86, 86, 87, 87, 88, 88, 89, 89, 90, 90, 91, 91, 92, 92, 93, 94, 95, 95, 96, 97, 97, 98, 98, 99, 99, 100, 101, 101, 102, 102, 103, 103, 104, 104, 105, 105, 106, 106, 107, 107, 108, 108, 109, 109, 110, 111, 111, 112, 112, 113, 113, 114, 114, 115, 116, 116, 117, 118, 118, 119, 120, 120, 121, 121, 122, 123, 123, 124, 124, 125, 125, 126, 126, 127, 127, 128, 128, 129, 129, 130, 131, 131, 132, 133, 133, 134, 134, 135, 136, 136, 137, 138, 138, 139, 139, 140, 140, 141, 141, 142, 142, 143, 143, 144, 144, 145, 145, 146, 146, 147, 148, 148, 149, 149, 150, 150, 151, 151, 152, 152, 153, 153, 154, 154, 155, 155, 156, 157, 157, 158, 159, 160, 161, 161, 162, 162, 163, 163, 164, 164, 165, 165, 166, 166, 167, 167, 168, 168, 169, 169, 170, 170, 171, 171, 172, 172, 173, 173, 174, 174, 175, 175, 176, 176, 177, 177)), .Label = c("1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16", "17", "18", "19", "20", "21", "22", "23", "24", "25", "26", "27", "28", "29", "30", "31", "32", "33", "34", "35", "36", "37", "38", "39", "40", "41", "42", "43", "44", "45", "46", "47", "48", "49", "50", "51", "52", "53", "54", "55", "56", "57", "58", "59", "60", "61", "62", "63", "64", "65", "66", "67", "68", "69", "70", "71", "72", "73", "74", "75", "76", "77", "78", "79", "80", "81", "82", "83", "84", "85", "86", "87", "88", "89", "90", "91", "92", "93", "94", "95", "96", "97", "98", "99", "100", "101", "102", "103", "104", "105", "106", "107", "108", "109", "110", "111", "112", "113", "114", "115", "116", "117", "118", "119", "120", "121", "122", "123", "124", "125", "126", "127", "128", "129", "130", "131", "132", "133", "134", "135", "136", "137", "138", "139", "140", "141", "142", "143", "144", "145", "146", "147", "148", "149", "150", "151", "152", "153", "154", "155", "156", "157", "158", "159", "160", "161", "162", "163", "164", "165", "166", "167", "168", "169", "170", "171", "172", "173", "174", "175", "176", "177"), class = "factor"), y = structure(as.integer(c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1)), .Label = c("0", "1"), class = "factor"), x = c(0.896492660264945, 0.896492660264945, 1.59446707642661, 1.59446707642661, -1.05008338359102, -1.05008338359102, 1.09348658068790, 1.09348658068790, 1.12507994528403, 1.12507994528403, 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