[R] lmer and mixed effects logistic regression
Rick Bilonick
rab45+ at pitt.edu
Mon Jun 19 16:14:14 CEST 2006
On Sun, 2006-06-18 at 13:58 +0200, Douglas Bates wrote:
> If I understand correctly Rick it trying to fit a model with random
> effects on a binary response when there are either 1 or 2 observations
> per group. I think that is very optimistic because there is so little
> information available per random effect (exactly 1 or 2 bits of
> information per random effect). I'm not surprised that there are
> difficulties in fitting such a model.
>
> Rick should try to monitor the iterations to see what is happening to
> the parameter estimates and perhaps should try the Laplace
> approximation to the log-likelihood. This is done by adding method =
> "Laplace" and control = list(msVerbose = TRUE) to the call to lmer.
> This causes the PQL iterations to be followed by direct optimization
> of the Laplace approximation. Because the problem is probably in the
> PQL iterations Rick may want to turn those off and do direct
> optimization of the Laplace approximation only. In that case the
> control argument should be list(usePQL=FALSE, msVerbose = TRUE)
>
>
> On 6/17/06, Spencer Graves <spencer.graves at pdf.com> wrote:
> >
> >
> > 'lmer' RETURNS AN ERROR WHEN SAS NLMIXED RETURNS AN ANSWER
> >
> > Like you, I would expect lmer to return an answer when SAS
> > NLMIXED does, and I'm concerned that it returns an error message instead.
> >
> > Your example is not self contained, and I've been unable to
> > get the result you got. This could occur for several reasons. First,
> > what do you get for "sessionInfo()"? I got the following:
> >
> > sessionInfo()
> > Version 2.3.1 (2006-06-01)
> > i386-pc-mingw32
> >
> > attached base packages:
> > [1] "methods" "stats" "graphics" "grDevices" "utils" "datasets"
> > [7] "base"
> >
> > other attached packages:
> > lme4 lattice Matrix
> > "0.995-2" "0.13-8" "0.995-10"
> >
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,
0.276572438987850,
0.276572438987850, 0.434273771509725, 0.434273771509725,
2.09093423436586, -0.622643744937437, -0.622643744937437,
-0.58706802345943, -0.58706802345943, 0.124446406100572,
0.124446406100572, 0.126570329770903, 0.126570329770903,
0.181261364281855, 1.64039692579746, 1.64039692579746,
-0.555474658863302,
-0.555474658863302, 0.47542479262234, 0.47542479262234,
-0.258656325934905,
-0.258656325934905, 1.64995458231394, 1.64995458231394,
0.696047363877697,
0.696047363877697, -1.46716889435175, -1.46716889435175,
1.03375122745992, 1.03375122745992, 0.790827457666109,
0.790827457666109,
-1.53194856629677, -1.53194856629677, -1.69389774615931,
-1.69389774615931, -0.811141970679076, -0.811141970679076,
-0.582289195201196, -0.582289195201196, 0.00789609469130197,
0.573390771916238, -1.45628378554133, -1.45628378554133,
-1.16079290490689, -1.16079290490689, -0.832646697841153,
-0.832646697841153, -0.241930427031072, -0.241930427031072,
-1.29353813430241, -1.29353813430241, -0.663794766050042,
-0.663794766050042, -0.961940551272396, -0.961940551272396,
1.59499805734419, 1.59499805734419, 0.47144243574048,
0.47144243574048,
0.952245656611064, 0.952245656611064, -0.304586175305754,
-0.304586175305754, 0.71463169599307, 0.71463169599307,
-1.32141463247548,
-0.0983000888251169, -0.0983000888251169, 0.440114561603134,
0.440114561603134, -1.75761545626916, -0.409985887445808,
-0.409985887445808, -0.847514163533447, -0.847514163533447,
-1.09229636653880, -1.09229636653880, -1.00415353422017,
-1.00415353422017, -1.63681729751924, -1.63681729751924,
-1.74354446195324, -1.74354446195324, -1.65460515825824,
-1.65460515825824, -1.30760912861834, -1.30760912861834,
-1.38300841891499, -1.38300841891499, -0.628750025489628,
-0.628750025489628, 0.323564250193864, 0.323564250193864,
-0.524412275184748, -0.524412275184748, -0.486181649118838,
-0.486181649118838, 0.142234266839580, 0.142234266839580,
-1.74965074250543, -1.74965074250543, -0.299010875671146,
2.01049062535218, 2.01049062535218, -1.18229763206896,
-1.18229763206896,
0.83304044061388, 0.83304044061388, -1.44539867673089,
-1.44539867673089,
-0.391136064871645, -0.391136064871645, -0.118477363693237,
-0.118477363693237, -1.73531425773072, -1.73531425773072,
0.748083493800747, -1.70717226909887, -1.70717226909887,
-0.210602552893726, -0.210602552893726, 0.681976369561778,
0.681976369561778, -0.0138741229295563, -0.0138741229295563,
-0.532111498489687, -0.532111498489687, -0.585740571165474,
-0.202106858212414, -0.202106858212414, -0.121663249198723,
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Rick B.
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