[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,
> 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,
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>
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