[R] lmer and mixed effects logistic regression

Spencer Graves spencer.graves at pdf.com
Sat Jun 24 06:38:11 CEST 2006


	  Permit me to try to repeat what I said earlier a little more clearly: 
  When the outcomes are constant for each subject, either all 0's or all 
1's, the maximum likelihood estimate of the between-subject variance in 
Inf.  Any software that returns a different answer is wrong. This is NOT 
a criticism of 'lmer' or SAS NLMIXED:  This is a sufficiently rare, 
extreme case that the software does not test for it and doesn't handle 
it well when it occurs.  Adding other explanatory variables to the model 
only makes this problem worse, because anything that will produce 
complete separation for each subject will produce this kind of 
instability.

	 Consider the following:

library(lme4)
DF <- data.frame(y=c(0,0, 0,1, 1,1),
                  Subj=rep(letters[1:3], each=2),
                  x=rep(c(-1, 1), 3))
fit1 <- lmer(y~1+(1|Subj), data=DF, family=binomial)

# 'lmer' works fine here, because the outcomes from
# 1 of the 3 subjects is not constant.

 > fit.x <- lmer(y~x+(1|Subj), data=DF, family=binomial)
Warning message:
IRLS iterations for PQL did not converge

	  The addition of 'x' to the model now allows complete separation for 
each subject.  We see this in the result:

Generalized linear mixed model fit using PQL
<snip>
Random effects:
  Groups Name        Variance   Std.Dev.
  Subj   (Intercept) 3.5357e+20 1.8803e+10
number of obs: 6, groups: Subj, 3

Estimated scale (compare to 1)  9.9414e-09

Fixed effects:
                Estimate  Std. Error    z value Pr(>|z|)
(Intercept) -5.4172e-05  1.0856e+10  -4.99e-15        1
x            8.6474e+01  2.7397e+07 3.1563e-06        1

	  Note that the subject variance is 3.5e20, the estimate for x is 86 
wit a standard error of 2.7e7.  All three of these numbers are reaching 
for Inf;  lmer quit before it got there.

	  Does this make any sense, or are we still misunderstanding one another?

	  Hope this helps.
	  Spencer Graves

Rick Bilonick wrote:
> 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,
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> ), class = "data.frame")
> 
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