Random vs fixed effects - LMM
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I'm going to start this as a new thread since my original question was buried.
Ryan, Best overview of LMM, i've found...thank you. My confusion stems from trying to equate the predicted values of a DV with ordinary regression where in a fixed effects model: predY = B0+Bx1+Bx2+error As your reference states- For fixed effects: If the SES parameter estimate is 2.5, then for a unit increase in SES the dependent variable (ex., test score) will increase 2.5 units - the same interpretation as in ordinary regression. So, we're good here. But for interpretation of the random effects: Intraclass correlation: interpreting random effects variance components. Let the variance component estimate for the random factor id (meaning school id) = 8.61. Let the variance component estimate for Residual = 39.15. In this null model, since the school variance component is 18% of the total of both variance components, we would say that the school effect accounts for 18% of the variance in math scores. So, this would not make sense to plug into the above equation. This is why i am confused by tables in manuscripts that put the coefficients for both the fixed and random variables all into one table and discuss their significance like they all represent coefficients that predict the value of a DV. Thanks Carol Carol, I suggest you take some time reading though the online documentation to which I have linked below. Write back if you have specific questions pertaining to fixed versus random effects. http://faculty.chass.ncsu.edu/garson/PA765/multilevel.htm Ryan On Fri, Sep 30, 2011 at 7:29 PM, Parise, Carol A. <PariseC@sutterhealth.org> wrote: > Hi all, > > I have a question on interpretation of the random versus fixed effects > coefficients that come out of a mixed model. > > Fixed effects are somewhat straightfoward. From what i think i > understand, the coefficient represents a change in the DV for a 1 unit > change in the variable - just like in a generalized linear model. > Statistical signficance for an IV is determine by a t-test testing > whether the coefficient is different from 0. if the CIs of the > coefficient don't cross zero, you're in business. > > For random effects, the output is labeled 'estimate' but the table is > "covariance parameter estimates" and tells you the percent of variance > accounted for by the variable of interest. Statistical significance is > tested using the Wald Z. > > My confusion lies when i read the results of publications that use > linear mixed models. The examples i've been going through tend to > discuss these factors separately, discussing percent of variance > accounted for by the random effects and change in DV for increases in > the IV for the fixed factors. > > However, in several publications, both the random and fixed effects > are included in a single table and state something like "Table 1 > presents the mixed effects model results estimating the simultaneous > effects of **random and fixed effects**". They then go on and > interpret them all the same just as if they were all fixed effects. > > I suspect these publications are correct (at least i would hope peer > review would have caught this type of error) and I am the one who is > missing something. > > Thanks in advance for enlightenment. > > Carol > > > > > ===================== To manage your subscription to SPSSX-L, send a message to LISTSERV@LISTSERV.UGA.EDU (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD |
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Carol,
There are two types of predicted values that are usually of interest upon fitting a linear mixed model. The first equation with which you are familiar produces population-averaged predicted values. These are calculated using the estimated fixed effects parameters only, i.e., y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk Note: These predicted values will obviously not vary across random subject effects. The second type of predicted values takes into account both the estimated fixed effects parameters and the empirical best linear unbiased predictors ("EBLUPs") of the random effects of each observation, i.e., y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk + uj_hat Note: These predicted values can vary across random subject effects. When discussing predicted values we should be clear as to whether we're referring to population-averaged or subject-specific predicted values. It's really difficult for me to determine whether a random effect has been misinterpreted without having the actual article available. Ryan On Mon, Oct 3, 2011 at 1:54 PM, parisec <[hidden email]> wrote: > I'm going to start this as a new thread since my original question was > buried. > > Ryan, > > Best overview of LMM, i've found...thank you. > > My confusion stems from trying to equate the predicted values of a DV with > ordinary regression where in a fixed effects model: > > predY = B0+Bx1+Bx2+error > > As your reference states- For fixed effects: > > If the SES parameter estimate is 2.5, then for a unit increase in SES the > dependent variable (ex., test score) will increase 2.5 units - the same > interpretation as in ordinary regression. So, we're good here. > > > But for interpretation of the random effects: > > Intraclass correlation: interpreting random effects variance components. Let > the variance component estimate for the random factor id (meaning school id) > = 8.61. Let the variance component estimate for Residual = 39.15. In this > null model, since the school variance component is 18% of the total of both > variance components, we would say that the school effect accounts for 18% of > the variance in math scores. > > > So, this would not make sense to plug into the above equation. This is why i > am confused by tables in manuscripts that put the coefficients for both the > fixed and random variables all into one table and discuss their significance > like they all represent coefficients that predict the value of a DV. > > Thanks > Carol > > > > > > > > > > > Carol, > > I suggest you take some time reading though the online documentation to > which I have linked below. Write back if you have specific questions > pertaining to fixed versus random effects. > > http://faculty.chass.ncsu.edu/garson/PA765/multilevel.htm > > Ryan > > On Fri, Sep 30, 2011 at 7:29 PM, Parise, Carol A. > <[hidden email]> wrote: >> Hi all, >> >> I have a question on interpretation of the random versus fixed effects >> coefficients that come out of a mixed model. >> >> Fixed effects are somewhat straightfoward. From what i think i >> understand, the coefficient represents a change in the DV for a 1 unit >> change in the variable - just like in a generalized linear model. >> Statistical signficance for an IV is determine by a t-test testing >> whether the coefficient is different from 0. if the CIs of the >> coefficient don't cross zero, you're in business. >> >> For random effects, the output is labeled 'estimate' but the table is >> "covariance parameter estimates" and tells you the percent of variance >> accounted for by the variable of interest. Statistical significance is >> tested using the Wald Z. >> >> My confusion lies when i read the results of publications that use >> linear mixed models. The examples i've been going through tend to >> discuss these factors separately, discussing percent of variance >> accounted for by the random effects and change in DV for increases in >> the IV for the fixed factors. >> >> However, in several publications, both the random and fixed effects >> are included in a single table and state something like "Table 1 >> presents the mixed effects model results estimating the simultaneous >> effects of **random and fixed effects**". They then go on and >> interpret them all the same just as if they were all fixed effects. >> >> I suspect these publications are correct (at least i would hope peer >> review would have caught this type of error) and I am the one who is >> missing something. >> >> Thanks in advance for enlightenment. >> >> Carol >> >> >> >> >> > > ===================== > To manage your subscription to SPSSX-L, send a message to > [hidden email] (not to SPSSX-L), with no body text except the > command. To leave the list, send the command SIGNOFF SPSSX-L For a list of > commands to manage subscriptions, send the command INFO REFCARD > > > -- > View this message in context: http://spssx-discussion.1045642.n5.nabble.com/Random-vs-fixed-effects-LMM-tp4865842p4865842.html > Sent from the SPSSX Discussion mailing list archive at Nabble.com. > > ===================== > To manage your subscription to SPSSX-L, send a message to > [hidden email] (not to SPSSX-L), with no body text except the > command. To leave the list, send the command > SIGNOFF SPSSX-L > For a list of commands to manage subscriptions, send the command > INFO REFCARD > ===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD |
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Carol,
The point I made about subject-specific predicted values piqued my curiosity as to whether the TEST subcommand in the MIXED procedure could be employed to make such predictions. The answer appears to be affirmative. BELOW my name is a self-contained example via SPSS syntax. The code also provides further explanation (using the COMPUTE function) as to how population-averaged and subject-specific predicted values can be obtained from a random intercept model with a single categorical predictor with 3 levels. I'm not entirely certain that my thinking is correct, but so far so good. I'll write back if I realize I've made an error. One final point--it's obvious by my redundant COMPUTE statements for the population-averaged predicted values that these values do not vary across subjects. Other points can be made about my use of terminology etc., but I'll stop for now since it's getting late. :) Best, Ryan -- Note that this data example came from the following website: http://ftp.sas.com/samples/A55235 SPSS syntax begins now: ***** Create data set *****. DATA LIST /observation 1-2 ingot 4 metal 6 pressure 8-11(1). BEGIN DATA 01 1 3 67.0 02 1 2 71.9 03 1 1 72.2 04 2 3 67.5 05 2 2 68.8 06 2 1 66.4 07 3 3 76.0 08 3 2 82.6 09 3 1 74.5 10 4 3 72.7 11 4 2 78.1 12 4 1 67.3 13 5 3 73.1 14 5 2 74.2 15 5 1 73.2 16 6 3 65.8 17 6 2 70.8 18 6 1 68.7 19 7 3 75.6 20 7 2 84.9 21 7 1 69.0 END DATA. ***** random intercept model *****. MIXED pressure BY metal ingot /FIXED=metal | SSTYPE(3) /METHOD=REML /PRINT=CORB COVB DESCRIPTIVES G LMATRIX R SOLUTION TESTCOV /RANDOM=INGOT /TEST 'metal1 | ingot=1' intercept 1 metal 1 0 0 | ingot 1 0 0 0 0 0 0 /TEST 'metal2 | ingot=1' intercept 1 metal 0 1 0 | ingot 1 0 0 0 0 0 0 /TEST 'metal3 | ingot=1' intercept 1 metal 0 0 1 | ingot 1 0 0 0 0 0 0 /TEST 'metal1 | ingot=2' intercept 1 metal 1 0 0 | ingot 0 1 0 0 0 0 0 /TEST 'metal2 | ingot=2' intercept 1 metal 0 1 0 | ingot 0 1 0 0 0 0 0 /TEST 'metal3 | ingot=2' intercept 1 metal 0 0 1 | ingot 0 1 0 0 0 0 0 /SAVE=FIXPRED PRED. COMPUTE eblups=PRED_1-FXPRED_1. COMPUTE b0 = 71.10000000000004. COMPUTE b1_1v3 = -0.9142857142857328. COMPUTE b1_2v3 = 4.79999999999998. EXECUTE. IF (ingot=1 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=1 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=1 and metal=3) yhat_population_avg = b0. IF (ingot=2 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=2 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=2 and metal=3) yhat_population_avg = b0. IF (ingot=3 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=3 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=3 and metal=3) yhat_population_avg = b0. IF (ingot=4 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=4 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=4 and metal=3) yhat_population_avg = b0. IF (ingot=5 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=5 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=5 and metal=3) yhat_population_avg = b0. IF (ingot=6 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=6 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=6 and metal=3) yhat_population_avg = b0. IF (ingot=7 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=7 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=7 and metal=3) yhat_population_avg = b0. EXECUTE. IF (ingot=1 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=1 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=1 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=2 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=2 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=2 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=3 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=3 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=3 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=4 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=4 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=4 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=5 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=5 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=5 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=6 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=6 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=6 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=7 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=7 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=7 and metal=3) yhat_subject_specific = b0 + eblups. EXECUTE. On Mon, Oct 3, 2011 at 8:08 PM, R B <[hidden email]> wrote: > Carol, > > There are two types of predicted values that are usually of interest > upon fitting a linear mixed model. The first equation with which you > are familiar produces population-averaged predicted values. These are > calculated using the estimated fixed effects parameters only, i.e., > > y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk > > Note: These predicted values will obviously not vary across random > subject effects. > > The second type of predicted values takes into account both the > estimated fixed effects parameters and the empirical best linear > unbiased predictors ("EBLUPs") of the random effects of each > observation, i.e., > > y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk + uj_hat > > Note: These predicted values can vary across random subject effects. > > When discussing predicted values we should be clear as to whether > we're referring to population-averaged or subject-specific predicted > values. > > It's really difficult for me to determine whether a random effect has > been misinterpreted without having the actual article available. > > Ryan > > On Mon, Oct 3, 2011 at 1:54 PM, parisec <[hidden email]> wrote: >> I'm going to start this as a new thread since my original question was >> buried. >> >> Ryan, >> >> Best overview of LMM, i've found...thank you. >> >> My confusion stems from trying to equate the predicted values of a DV with >> ordinary regression where in a fixed effects model: >> >> predY = B0+Bx1+Bx2+error >> >> As your reference states- For fixed effects: >> >> If the SES parameter estimate is 2.5, then for a unit increase in SES the >> dependent variable (ex., test score) will increase 2.5 units - the same >> interpretation as in ordinary regression. So, we're good here. >> >> >> But for interpretation of the random effects: >> >> Intraclass correlation: interpreting random effects variance components. Let >> the variance component estimate for the random factor id (meaning school id) >> = 8.61. Let the variance component estimate for Residual = 39.15. In this >> null model, since the school variance component is 18% of the total of both >> variance components, we would say that the school effect accounts for 18% of >> the variance in math scores. >> >> >> So, this would not make sense to plug into the above equation. This is why i >> am confused by tables in manuscripts that put the coefficients for both the >> fixed and random variables all into one table and discuss their significance >> like they all represent coefficients that predict the value of a DV. >> >> Thanks >> Carol >> >> >> >> >> >> >> >> >> >> >> Carol, >> >> I suggest you take some time reading though the online documentation to >> which I have linked below. Write back if you have specific questions >> pertaining to fixed versus random effects. >> >> http://faculty.chass.ncsu.edu/garson/PA765/multilevel.htm >> >> Ryan >> >> On Fri, Sep 30, 2011 at 7:29 PM, Parise, Carol A. >> <[hidden email]> wrote: >>> Hi all, >>> >>> I have a question on interpretation of the random versus fixed effects >>> coefficients that come out of a mixed model. >>> >>> Fixed effects are somewhat straightfoward. From what i think i >>> understand, the coefficient represents a change in the DV for a 1 unit >>> change in the variable - just like in a generalized linear model. >>> Statistical signficance for an IV is determine by a t-test testing >>> whether the coefficient is different from 0. if the CIs of the >>> coefficient don't cross zero, you're in business. >>> >>> For random effects, the output is labeled 'estimate' but the table is >>> "covariance parameter estimates" and tells you the percent of variance >>> accounted for by the variable of interest. Statistical significance is >>> tested using the Wald Z. >>> >>> My confusion lies when i read the results of publications that use >>> linear mixed models. The examples i've been going through tend to >>> discuss these factors separately, discussing percent of variance >>> accounted for by the random effects and change in DV for increases in >>> the IV for the fixed factors. >>> >>> However, in several publications, both the random and fixed effects >>> are included in a single table and state something like "Table 1 >>> presents the mixed effects model results estimating the simultaneous >>> effects of **random and fixed effects**". They then go on and >>> interpret them all the same just as if they were all fixed effects. >>> >>> I suspect these publications are correct (at least i would hope peer >>> review would have caught this type of error) and I am the one who is >>> missing something. >>> >>> Thanks in advance for enlightenment. >>> >>> Carol >>> >>> >>> >>> >>> >> >> ===================== >> To manage your subscription to SPSSX-L, send a message to >> [hidden email] (not to SPSSX-L), with no body text except the >> command. To leave the list, send the command SIGNOFF SPSSX-L For a list of >> commands to manage subscriptions, send the command INFO REFCARD >> >> >> -- >> View this message in context: http://spssx-discussion.1045642.n5.nabble.com/Random-vs-fixed-effects-LMM-tp4865842p4865842.html >> Sent from the SPSSX Discussion mailing list archive at Nabble.com. >> >> ===================== >> To manage your subscription to SPSSX-L, send a message to >> [hidden email] (not to SPSSX-L), with no body text except the >> command. To leave the list, send the command >> SIGNOFF SPSSX-L >> For a list of commands to manage subscriptions, send the command >> INFO REFCARD >> > ===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD |
Re: Random vs fixed effects - LMM
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Very useful post Ryan!! Looks like these can be reduced to: DO IF METAL=1. + COMPUTE yhat_population_avg = b0 + b1_1v3. ELSE IF METAL=2. + COMPUTE yhat_population_avg = b0 + b1_2v3. ELSE. + COMPUTE yhat_population_avg = b0. END IF. COMPUTE yhat_subject_specific=yhat_population_avg + eblups. --- OR even more succinctly: COMPUTE yhat_population_avg =b0 + (METAL=1)*b1_1v3 + (METAL=2)*b1_2v3. COMPUTE yhat_subject_specific=yhat_population_avg + eblups. Makes total sense!! David
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In reply to this post by Ryan
Ryan,
Great information! I've been playing around with your example. Something that still gnaws at me is what values are being used for the EBLUPs. You obtained them in your example by subtracting the predicted values - fixed predicted. This means that to compute the predicted values that spss saves, an EBLUP has to be computed for each of the random variables. Do you know if this value can be saved? Yes, I know i am getting hung up on a minor detail but somehow, looking at predicted values versus actual values of individual cases really clears up the mystery behind these models. On the topic of how to deal with random variables in publications, i've been looking through several pubmed 'free articles' that use mixed linear models. It appears as if most publications just mention the variables that were included as random variables and never go any further since the fixed effects are the variables of interest. I have a model where the fixed effects are the variables of interest but the random effects should be evaluated. I'm thinking I can't go wrong if i stick with discussing the variance accounted for by these variables. Any thoughts on this? Thank you. Carol -----Original Message----- From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of R B Sent: Monday, October 03, 2011 7:44 PM To: [hidden email] Subject: Re: Random vs fixed effects - LMM Carol, The point I made about subject-specific predicted values piqued my curiosity as to whether the TEST subcommand in the MIXED procedure could be employed to make such predictions. The answer appears to be affirmative. BELOW my name is a self-contained example via SPSS syntax. The code also provides further explanation (using the COMPUTE function) as to how population-averaged and subject-specific predicted values can be obtained from a random intercept model with a single categorical predictor with 3 levels. I'm not entirely certain that my thinking is correct, but so far so good. I'll write back if I realize I've made an error. One final point--it's obvious by my redundant COMPUTE statements for the population-averaged predicted values that these values do not vary across subjects. Other points can be made about my use of terminology etc., but I'll stop for now since it's getting late. :) Best, Ryan -- Note that this data example came from the following website: http://ftp.sas.com/samples/A55235 SPSS syntax begins now: ***** Create data set *****. DATA LIST /observation 1-2 ingot 4 metal 6 pressure 8-11(1). BEGIN DATA 01 1 3 67.0 02 1 2 71.9 03 1 1 72.2 04 2 3 67.5 05 2 2 68.8 06 2 1 66.4 07 3 3 76.0 08 3 2 82.6 09 3 1 74.5 10 4 3 72.7 11 4 2 78.1 12 4 1 67.3 13 5 3 73.1 14 5 2 74.2 15 5 1 73.2 16 6 3 65.8 17 6 2 70.8 18 6 1 68.7 19 7 3 75.6 20 7 2 84.9 21 7 1 69.0 END DATA. ***** random intercept model *****. MIXED pressure BY metal ingot /FIXED=metal | SSTYPE(3) /METHOD=REML /PRINT=CORB COVB DESCRIPTIVES G LMATRIX R SOLUTION TESTCOV /RANDOM=INGOT /TEST 'metal1 | ingot=1' intercept 1 metal 1 0 0 | ingot 1 0 0 0 0 0 0 /TEST 'metal2 | ingot=1' intercept 1 metal 0 1 0 | ingot 1 0 0 0 0 0 0 /TEST 'metal3 | ingot=1' intercept 1 metal 0 0 1 | ingot 1 0 0 0 0 0 0 /TEST 'metal1 | ingot=2' intercept 1 metal 1 0 0 | ingot 0 1 0 0 0 0 0 /TEST 'metal2 | ingot=2' intercept 1 metal 0 1 0 | ingot 0 1 0 0 0 0 0 /TEST 'metal3 | ingot=2' intercept 1 metal 0 0 1 | ingot 0 1 0 0 0 0 0 /SAVE=FIXPRED PRED. COMPUTE eblups=PRED_1-FXPRED_1. COMPUTE b0 = 71.10000000000004. COMPUTE b1_1v3 = -0.9142857142857328. COMPUTE b1_2v3 = 4.79999999999998. EXECUTE. IF (ingot=1 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=1 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=1 and metal=3) yhat_population_avg = b0. IF (ingot=2 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=2 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=2 and metal=3) yhat_population_avg = b0. IF (ingot=3 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=3 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=3 and metal=3) yhat_population_avg = b0. IF (ingot=4 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=4 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=4 and metal=3) yhat_population_avg = b0. IF (ingot=5 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=5 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=5 and metal=3) yhat_population_avg = b0. IF (ingot=6 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=6 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=6 and metal=3) yhat_population_avg = b0. IF (ingot=7 and metal=1) yhat_population_avg = b0 + b1_1v3. IF (ingot=7 and metal=2) yhat_population_avg = b0 + b1_2v3. IF (ingot=7 and metal=3) yhat_population_avg = b0. EXECUTE. IF (ingot=1 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=1 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=1 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=2 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=2 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=2 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=3 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=3 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=3 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=4 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=4 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=4 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=5 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=5 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=5 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=6 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=6 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=6 and metal=3) yhat_subject_specific = b0 + eblups. IF (ingot=7 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. IF (ingot=7 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. IF (ingot=7 and metal=3) yhat_subject_specific = b0 + eblups. EXECUTE. On Mon, Oct 3, 2011 at 8:08 PM, R B <[hidden email]> wrote: > Carol, > > There are two types of predicted values that are usually of interest > upon fitting a linear mixed model. The first equation with which you > are familiar produces population-averaged predicted values. These are > calculated using the estimated fixed effects parameters only, i.e., > > y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk > > Note: These predicted values will obviously not vary across random > subject effects. > > The second type of predicted values takes into account both the > estimated fixed effects parameters and the empirical best linear > unbiased predictors ("EBLUPs") of the random effects of each > observation, i.e., > > y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk + uj_hat > > Note: These predicted values can vary across random subject effects. > > When discussing predicted values we should be clear as to whether > we're referring to population-averaged or subject-specific predicted > values. > > It's really difficult for me to determine whether a random effect has > been misinterpreted without having the actual article available. > > Ryan > > On Mon, Oct 3, 2011 at 1:54 PM, parisec <[hidden email]> wrote: >> I'm going to start this as a new thread since my original question >> was buried. >> >> Ryan, >> >> Best overview of LMM, i've found...thank you. >> >> My confusion stems from trying to equate the predicted values of a DV >> with ordinary regression where in a fixed effects model: >> >> predY = B0+Bx1+Bx2+error >> >> As your reference states- For fixed effects: >> >> If the SES parameter estimate is 2.5, then for a unit increase in SES >> the dependent variable (ex., test score) will increase 2.5 units - >> the same interpretation as in ordinary regression. So, we're good here. >> >> >> But for interpretation of the random effects: >> >> Intraclass correlation: interpreting random effects variance >> components. Let the variance component estimate for the random factor >> id (meaning school id) = 8.61. Let the variance component estimate >> for Residual = 39.15. In this null model, since the school variance >> component is 18% of the total of both variance components, we would >> say that the school effect accounts for 18% of the variance in math scores. >> >> >> So, this would not make sense to plug into the above equation. This >> is why i am confused by tables in manuscripts that put the >> coefficients for both the fixed and random variables all into one >> table and discuss their significance like they all represent coefficients that predict the value of a DV. >> >> Thanks >> Carol >> >> >> >> >> >> >> >> >> >> >> Carol, >> >> I suggest you take some time reading though the online documentation >> to which I have linked below. Write back if you have specific >> questions pertaining to fixed versus random effects. >> >> http://faculty.chass.ncsu.edu/garson/PA765/multilevel.htm >> >> Ryan >> >> On Fri, Sep 30, 2011 at 7:29 PM, Parise, Carol A. >> <[hidden email]> wrote: >>> Hi all, >>> >>> I have a question on interpretation of the random versus fixed >>> effects coefficients that come out of a mixed model. >>> >>> Fixed effects are somewhat straightfoward. From what i think i >>> understand, the coefficient represents a change in the DV for a 1 >>> unit change in the variable - just like in a generalized linear model. >>> Statistical signficance for an IV is determine by a t-test testing >>> whether the coefficient is different from 0. if the CIs of the >>> coefficient don't cross zero, you're in business. >>> >>> For random effects, the output is labeled 'estimate' but the table >>> is "covariance parameter estimates" and tells you the percent of >>> variance accounted for by the variable of interest. Statistical >>> significance is tested using the Wald Z. >>> >>> My confusion lies when i read the results of publications that use >>> linear mixed models. The examples i've been going through tend to >>> discuss these factors separately, discussing percent of variance >>> accounted for by the random effects and change in DV for increases >>> in the IV for the fixed factors. >>> >>> However, in several publications, both the random and fixed effects >>> are included in a single table and state something like "Table 1 >>> presents the mixed effects model results estimating the simultaneous >>> effects of **random and fixed effects**". They then go on and >>> interpret them all the same just as if they were all fixed effects. >>> >>> I suspect these publications are correct (at least i would hope peer >>> review would have caught this type of error) and I am the one who >>> is missing something. >>> >>> Thanks in advance for enlightenment. >>> >>> Carol >>> >>> >>> >>> >>> >> >> ===================== >> To manage your subscription to SPSSX-L, send a message to >> [hidden email] (not to SPSSX-L), with no body text except >> the command. To leave the list, send the command SIGNOFF SPSSX-L For >> a list of commands to manage subscriptions, send the command INFO >> REFCARD >> >> >> -- >> View this message in context: >> http://spssx-discussion.1045642.n5.nabble.com/Random-vs-fixed-effects >> -LMM-tp4865842p4865842.html Sent from the SPSSX Discussion mailing >> list archive at Nabble.com. >> >> ===================== >> To manage your subscription to SPSSX-L, send a message to >> [hidden email] (not to SPSSX-L), with no body text except >> the command. To leave the list, send the command SIGNOFF SPSSX-L For >> a list of commands to manage subscriptions, send the command INFO >> REFCARD >> > ===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD ===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD |
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Carol,
One way to obtain the EBLUPS of the random effects of the observations for the example provided previously would be to incorporate the following TEST statements: /TEST 'ingot 1' | ingot 1 0 0 0 0 0 0 /TEST 'ingot 2' | ingot 0 1 0 0 0 0 0 /TEST 'ingot 3' | ingot 0 0 1 0 0 0 0 /TEST 'ingot 4' | ingot 0 0 0 1 0 0 0 /TEST 'ingot 5' | ingot 0 0 0 0 1 0 0 /TEST 'ingot 6' | ingot 0 0 0 0 0 1 0 /TEST 'ingot 7' | ingot 0 0 0 0 0 0 1 I do not know of another way to directly "SAVE" the EBLUPS of the random effects of each observation from the MIXED procedure in SPSS. You might be curious about the equation used to obtain these EBLUPS: compute EBLUP_subject1 = (11.447777777777192 / (11.447777777777192 + 1/3 * (10.371587301588878))) * (70.36666666666667 - 72.39523809523811). compute EBLUP_subject2 = (11.447777777777192 / (11.447777777777192 + 1/3 * (10.371587301588878))) * (67.56666666666666 - 72.39523809523811). compute EBLUP_subject3 = (11.447777777777192 / (11.447777777777192 + 1/3 * (10.371587301588878))) * (77.70000000000000 - 72.39523809523811). compute EBLUP_subject4 = (11.447777777777192 / (11.447777777777192 + 1/3 * (10.371587301588878))) * (72.70000000000000 - 72.39523809523811). compute EBLUP_subject5 = (11.447777777777192 / (11.447777777777192 + 1/3 * (10.371587301588878))) * (73.50000000000000 - 72.39523809523811). compute EBLUP_subject6 = (11.447777777777192 / (11.447777777777192 + 1/3 * (10.371587301588878))) * (68.43333333333334 - 72.39523809523811). compute EBLUP_subject7 = (11.447777777777192 / (11.447777777777192 + 1/3 * (10.371587301588878))) * (76.50000000000000 - 72.39523809523811). execute. The formula for the equations provided in the COMPUTE statements above is: [sigma^2_j / (sigma^2_j + 1/3*sigma)] * (ybar_.j - mu_hat) where sigma^2_j = between subject variance sigma = residual variance ybar_.j = mean of the jth subject mu_hat = grand mean Note that when I use the terms "sigma^2_j" and "sigma", I am referring to the ESTIMATED variances which are provided in the "Estimates of Covariance Parameters" Table in the MIXED output. I agree that even if the focus of an article is on the fixed effects estimates, some space should be spent interpreting the random effects estimates. Ryan On Tue, Oct 4, 2011 at 1:35 PM, Parise, Carol A. <[hidden email]> wrote: > Ryan, > > Great information! I've been playing around with your example. Something that still gnaws at me is what values are being used for the EBLUPs. You obtained them in your example by subtracting the predicted values - fixed predicted. This means that to compute the predicted values that spss saves, an EBLUP has to be computed for each of the random variables. Do you know if this value can be saved? Yes, I know i am getting hung up on a minor detail but somehow, looking at predicted values versus actual values of individual cases really clears up the mystery behind these models. > > On the topic of how to deal with random variables in publications, i've been looking through several pubmed 'free articles' that use mixed linear models. It appears as if most publications just mention the variables that were included as random variables and never go any further since the fixed effects are the variables of interest. > > I have a model where the fixed effects are the variables of interest but the random effects should be evaluated. I'm thinking I can't go wrong if i stick with discussing the variance accounted for by these variables. Any thoughts on this? > > Thank you. > Carol > > -----Original Message----- > From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of R B > Sent: Monday, October 03, 2011 7:44 PM > To: [hidden email] > Subject: Re: Random vs fixed effects - LMM > > Carol, > > The point I made about subject-specific predicted values piqued my curiosity as to whether the TEST subcommand in the MIXED procedure could be employed to make such predictions. The answer appears to be affirmative. BELOW my name is a self-contained example via SPSS syntax. The code also provides further explanation (using the COMPUTE > function) as to how population-averaged and subject-specific predicted values can be obtained from a random intercept model with a single categorical predictor with 3 levels. I'm not entirely certain that my thinking is correct, but so far so good. I'll write back if I realize I've made an error. One final point--it's obvious by my redundant COMPUTE statements for the population-averaged predicted values that these values do not vary across subjects. Other points can be made about my use of terminology etc., but I'll stop for now since it's getting late. :) > > Best, > > Ryan > -- > > Note that this data example came from the following website: > > http://ftp.sas.com/samples/A55235 > > SPSS syntax begins now: > > ***** Create data set *****. > DATA LIST /observation 1-2 ingot 4 metal 6 pressure 8-11(1). > BEGIN DATA > 01 1 3 67.0 > 02 1 2 71.9 > 03 1 1 72.2 > 04 2 3 67.5 > 05 2 2 68.8 > 06 2 1 66.4 > 07 3 3 76.0 > 08 3 2 82.6 > 09 3 1 74.5 > 10 4 3 72.7 > 11 4 2 78.1 > 12 4 1 67.3 > 13 5 3 73.1 > 14 5 2 74.2 > 15 5 1 73.2 > 16 6 3 65.8 > 17 6 2 70.8 > 18 6 1 68.7 > 19 7 3 75.6 > 20 7 2 84.9 > 21 7 1 69.0 > END DATA. > > ***** random intercept model *****. > MIXED pressure BY metal ingot > /FIXED=metal | SSTYPE(3) > /METHOD=REML > /PRINT=CORB COVB DESCRIPTIVES G LMATRIX R SOLUTION TESTCOV > /RANDOM=INGOT > /TEST 'metal1 | ingot=1' intercept 1 metal 1 0 0 | ingot 1 0 0 0 0 0 0 > /TEST 'metal2 | ingot=1' intercept 1 metal 0 1 0 | ingot 1 0 0 0 0 0 0 > /TEST 'metal3 | ingot=1' intercept 1 metal 0 0 1 | ingot 1 0 0 0 0 0 0 > /TEST 'metal1 | ingot=2' intercept 1 metal 1 0 0 | ingot 0 1 0 0 0 0 0 > /TEST 'metal2 | ingot=2' intercept 1 metal 0 1 0 | ingot 0 1 0 0 0 0 0 > /TEST 'metal3 | ingot=2' intercept 1 metal 0 0 1 | ingot 0 1 0 0 0 0 0 > /SAVE=FIXPRED PRED. > > COMPUTE eblups=PRED_1-FXPRED_1. > COMPUTE b0 = 71.10000000000004. > COMPUTE b1_1v3 = -0.9142857142857328. > COMPUTE b1_2v3 = 4.79999999999998. > EXECUTE. > > IF (ingot=1 and metal=1) yhat_population_avg = b0 + b1_1v3. > IF (ingot=1 and metal=2) yhat_population_avg = b0 + b1_2v3. > IF (ingot=1 and metal=3) yhat_population_avg = b0. > IF (ingot=2 and metal=1) yhat_population_avg = b0 + b1_1v3. > IF (ingot=2 and metal=2) yhat_population_avg = b0 + b1_2v3. > IF (ingot=2 and metal=3) yhat_population_avg = b0. > IF (ingot=3 and metal=1) yhat_population_avg = b0 + b1_1v3. > IF (ingot=3 and metal=2) yhat_population_avg = b0 + b1_2v3. > IF (ingot=3 and metal=3) yhat_population_avg = b0. > IF (ingot=4 and metal=1) yhat_population_avg = b0 + b1_1v3. > IF (ingot=4 and metal=2) yhat_population_avg = b0 + b1_2v3. > IF (ingot=4 and metal=3) yhat_population_avg = b0. > IF (ingot=5 and metal=1) yhat_population_avg = b0 + b1_1v3. > IF (ingot=5 and metal=2) yhat_population_avg = b0 + b1_2v3. > IF (ingot=5 and metal=3) yhat_population_avg = b0. > IF (ingot=6 and metal=1) yhat_population_avg = b0 + b1_1v3. > IF (ingot=6 and metal=2) yhat_population_avg = b0 + b1_2v3. > IF (ingot=6 and metal=3) yhat_population_avg = b0. > IF (ingot=7 and metal=1) yhat_population_avg = b0 + b1_1v3. > IF (ingot=7 and metal=2) yhat_population_avg = b0 + b1_2v3. > IF (ingot=7 and metal=3) yhat_population_avg = b0. > EXECUTE. > > IF (ingot=1 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. > IF (ingot=1 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. > IF (ingot=1 and metal=3) yhat_subject_specific = b0 + eblups. > IF (ingot=2 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. > IF (ingot=2 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. > IF (ingot=2 and metal=3) yhat_subject_specific = b0 + eblups. > IF (ingot=3 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. > IF (ingot=3 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. > IF (ingot=3 and metal=3) yhat_subject_specific = b0 + eblups. > IF (ingot=4 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. > IF (ingot=4 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. > IF (ingot=4 and metal=3) yhat_subject_specific = b0 + eblups. > IF (ingot=5 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. > IF (ingot=5 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. > IF (ingot=5 and metal=3) yhat_subject_specific = b0 + eblups. > IF (ingot=6 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. > IF (ingot=6 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. > IF (ingot=6 and metal=3) yhat_subject_specific = b0 + eblups. > IF (ingot=7 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. > IF (ingot=7 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. > IF (ingot=7 and metal=3) yhat_subject_specific = b0 + eblups. > EXECUTE. > > On Mon, Oct 3, 2011 at 8:08 PM, R B <[hidden email]> wrote: >> Carol, >> >> There are two types of predicted values that are usually of interest >> upon fitting a linear mixed model. The first equation with which you >> are familiar produces population-averaged predicted values. These are >> calculated using the estimated fixed effects parameters only, i.e., >> >> y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk >> >> Note: These predicted values will obviously not vary across random >> subject effects. >> >> The second type of predicted values takes into account both the >> estimated fixed effects parameters and the empirical best linear >> unbiased predictors ("EBLUPs") of the random effects of each >> observation, i.e., >> >> y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk + uj_hat >> >> Note: These predicted values can vary across random subject effects. >> >> When discussing predicted values we should be clear as to whether >> we're referring to population-averaged or subject-specific predicted >> values. >> >> It's really difficult for me to determine whether a random effect has >> been misinterpreted without having the actual article available. >> >> Ryan >> >> On Mon, Oct 3, 2011 at 1:54 PM, parisec <[hidden email]> wrote: >>> I'm going to start this as a new thread since my original question >>> was buried. >>> >>> Ryan, >>> >>> Best overview of LMM, i've found...thank you. >>> >>> My confusion stems from trying to equate the predicted values of a DV >>> with ordinary regression where in a fixed effects model: >>> >>> predY = B0+Bx1+Bx2+error >>> >>> As your reference states- For fixed effects: >>> >>> If the SES parameter estimate is 2.5, then for a unit increase in SES >>> the dependent variable (ex., test score) will increase 2.5 units - >>> the same interpretation as in ordinary regression. So, we're good here. >>> >>> >>> But for interpretation of the random effects: >>> >>> Intraclass correlation: interpreting random effects variance >>> components. Let the variance component estimate for the random factor >>> id (meaning school id) = 8.61. Let the variance component estimate >>> for Residual = 39.15. In this null model, since the school variance >>> component is 18% of the total of both variance components, we would >>> say that the school effect accounts for 18% of the variance in math scores. >>> >>> >>> So, this would not make sense to plug into the above equation. This >>> is why i am confused by tables in manuscripts that put the >>> coefficients for both the fixed and random variables all into one >>> table and discuss their significance like they all represent coefficients that predict the value of a DV. >>> >>> Thanks >>> Carol >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> Carol, >>> >>> I suggest you take some time reading though the online documentation >>> to which I have linked below. Write back if you have specific >>> questions pertaining to fixed versus random effects. >>> >>> http://faculty.chass.ncsu.edu/garson/PA765/multilevel.htm >>> >>> Ryan >>> >>> On Fri, Sep 30, 2011 at 7:29 PM, Parise, Carol A. >>> <[hidden email]> wrote: >>>> Hi all, >>>> >>>> I have a question on interpretation of the random versus fixed >>>> effects coefficients that come out of a mixed model. >>>> >>>> Fixed effects are somewhat straightfoward. From what i think i >>>> understand, the coefficient represents a change in the DV for a 1 >>>> unit change in the variable - just like in a generalized linear model. >>>> Statistical signficance for an IV is determine by a t-test testing >>>> whether the coefficient is different from 0. if the CIs of the >>>> coefficient don't cross zero, you're in business. >>>> >>>> For random effects, the output is labeled 'estimate' but the table >>>> is "covariance parameter estimates" and tells you the percent of >>>> variance accounted for by the variable of interest. Statistical >>>> significance is tested using the Wald Z. >>>> >>>> My confusion lies when i read the results of publications that use >>>> linear mixed models. The examples i've been going through tend to >>>> discuss these factors separately, discussing percent of variance >>>> accounted for by the random effects and change in DV for increases >>>> in the IV for the fixed factors. >>>> >>>> However, in several publications, both the random and fixed effects >>>> are included in a single table and state something like "Table 1 >>>> presents the mixed effects model results estimating the simultaneous >>>> effects of **random and fixed effects**". They then go on and >>>> interpret them all the same just as if they were all fixed effects. >>>> >>>> I suspect these publications are correct (at least i would hope peer >>>> review would have caught this type of error) and I am the one who >>>> is missing something. >>>> >>>> Thanks in advance for enlightenment. >>>> >>>> Carol >>>> >>>> >>>> >>>> >>>> >>> >>> ===================== >>> To manage your subscription to SPSSX-L, send a message to >>> [hidden email] (not to SPSSX-L), with no body text except >>> the command. To leave the list, send the command SIGNOFF SPSSX-L For >>> a list of commands to manage subscriptions, send the command INFO >>> REFCARD >>> >>> >>> -- >>> View this message in context: >>> http://spssx-discussion.1045642.n5.nabble.com/Random-vs-fixed-effects >>> -LMM-tp4865842p4865842.html Sent from the SPSSX Discussion mailing >>> list archive at Nabble.com. >>> >>> ===================== >>> To manage your subscription to SPSSX-L, send a message to >>> [hidden email] (not to SPSSX-L), with no body text except >>> the command. To leave the list, send the command SIGNOFF SPSSX-L For >>> a list of commands to manage subscriptions, send the command INFO >>> REFCARD >>> >> > > ===================== > To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD > ===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD |
Re: Random vs fixed effects - LMM
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Administrator
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Sorry, couldn't help myself ;-).
COMPUTE #SigHatj=11.447777777777192. COMPUTE #SigHat =10.371587301588878. COMPUTE #MuHat =72.39523809523811. DO REPEAT EBLUP=EBLUP_S1 TO EBLUP_S7 /Y_BarJ=70.36666666666667 67.56666666666666 77.7 72.7 73.5 68.43333333333334 76.5 . COMPUTE EBLUP=#SigHatj *(Y_BarJ-#MuHat) /(#SigHatj + 1/3 * #SigHat) . END REPEAT.
Please reply to the list and not to my personal email.
Those desiring my consulting or training services please feel free to email me. --- "Nolite dare sanctum canibus neque mittatis margaritas vestras ante porcos ne forte conculcent eas pedibus suis." Cum es damnatorum possederunt porcos iens ut salire off sanguinum cliff in abyssum?" |
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In reply to this post by Ryan
Carol,
I just noticed that the I used the term "sigma" instead of "sigma^2" in the EBLUPs formula by mistake. So, it should be: [sigma^2_j / (sigma^2_j + 1/3*sigma^2)] * (ybar_.j - mu_hat) Also, keep in mind that the variances are ESTIMATED, so technically they should have a "hat" at the end of the terms. Just didn't want to confuse matters given the superscript and subscripts already included. Also, special thanks to David for providing more efficient code. Ryan On Tue, Oct 4, 2011 at 9:22 PM, R B <[hidden email]> wrote: > Carol, > > One way to obtain the EBLUPS of the random effects of the observations > for the example provided previously would be to incorporate the > following TEST statements: > > /TEST 'ingot 1' | ingot 1 0 0 0 0 0 0 > /TEST 'ingot 2' | ingot 0 1 0 0 0 0 0 > /TEST 'ingot 3' | ingot 0 0 1 0 0 0 0 > /TEST 'ingot 4' | ingot 0 0 0 1 0 0 0 > /TEST 'ingot 5' | ingot 0 0 0 0 1 0 0 > /TEST 'ingot 6' | ingot 0 0 0 0 0 1 0 > /TEST 'ingot 7' | ingot 0 0 0 0 0 0 1 > > I do not know of another way to directly "SAVE" the EBLUPS of the > random effects of each observation from the MIXED procedure in SPSS. > > You might be curious about the equation used to obtain these EBLUPS: > > compute EBLUP_subject1 = (11.447777777777192 / (11.447777777777192 + > 1/3 * (10.371587301588878))) * (70.36666666666667 - > 72.39523809523811). > compute EBLUP_subject2 = (11.447777777777192 / (11.447777777777192 + > 1/3 * (10.371587301588878))) * (67.56666666666666 - > 72.39523809523811). > compute EBLUP_subject3 = (11.447777777777192 / (11.447777777777192 + > 1/3 * (10.371587301588878))) * (77.70000000000000 - > 72.39523809523811). > compute EBLUP_subject4 = (11.447777777777192 / (11.447777777777192 + > 1/3 * (10.371587301588878))) * (72.70000000000000 - > 72.39523809523811). > compute EBLUP_subject5 = (11.447777777777192 / (11.447777777777192 + > 1/3 * (10.371587301588878))) * (73.50000000000000 - > 72.39523809523811). > compute EBLUP_subject6 = (11.447777777777192 / (11.447777777777192 + > 1/3 * (10.371587301588878))) * (68.43333333333334 - > 72.39523809523811). > compute EBLUP_subject7 = (11.447777777777192 / (11.447777777777192 + > 1/3 * (10.371587301588878))) * (76.50000000000000 - > 72.39523809523811). > execute. > > The formula for the equations provided in the COMPUTE statements above is: > > [sigma^2_j / (sigma^2_j + 1/3*sigma)] * (ybar_.j - mu_hat) > > where > > sigma^2_j = between subject variance > sigma = residual variance > ybar_.j = mean of the jth subject > mu_hat = grand mean > > Note that when I use the terms "sigma^2_j" and "sigma", I am referring > to the ESTIMATED variances which are provided in the "Estimates of > Covariance Parameters" Table in the MIXED output. > > I agree that even if the focus of an article is on the fixed effects > estimates, some space should be spent interpreting the random effects > estimates. > > Ryan > > On Tue, Oct 4, 2011 at 1:35 PM, Parise, Carol A. > <[hidden email]> wrote: >> Ryan, >> >> Great information! I've been playing around with your example. Something that still gnaws at me is what values are being used for the EBLUPs. You obtained them in your example by subtracting the predicted values - fixed predicted. This means that to compute the predicted values that spss saves, an EBLUP has to be computed for each of the random variables. Do you know if this value can be saved? Yes, I know i am getting hung up on a minor detail but somehow, looking at predicted values versus actual values of individual cases really clears up the mystery behind these models. >> >> On the topic of how to deal with random variables in publications, i've been looking through several pubmed 'free articles' that use mixed linear models. It appears as if most publications just mention the variables that were included as random variables and never go any further since the fixed effects are the variables of interest. >> >> I have a model where the fixed effects are the variables of interest but the random effects should be evaluated. I'm thinking I can't go wrong if i stick with discussing the variance accounted for by these variables. Any thoughts on this? >> >> Thank you. >> Carol >> >> -----Original Message----- >> From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of R B >> Sent: Monday, October 03, 2011 7:44 PM >> To: [hidden email] >> Subject: Re: Random vs fixed effects - LMM >> >> Carol, >> >> The point I made about subject-specific predicted values piqued my curiosity as to whether the TEST subcommand in the MIXED procedure could be employed to make such predictions. The answer appears to be affirmative. BELOW my name is a self-contained example via SPSS syntax. The code also provides further explanation (using the COMPUTE >> function) as to how population-averaged and subject-specific predicted values can be obtained from a random intercept model with a single categorical predictor with 3 levels. I'm not entirely certain that my thinking is correct, but so far so good. I'll write back if I realize I've made an error. One final point--it's obvious by my redundant COMPUTE statements for the population-averaged predicted values that these values do not vary across subjects. Other points can be made about my use of terminology etc., but I'll stop for now since it's getting late. :) >> >> Best, >> >> Ryan >> -- >> >> Note that this data example came from the following website: >> >> http://ftp.sas.com/samples/A55235 >> >> SPSS syntax begins now: >> >> ***** Create data set *****. >> DATA LIST /observation 1-2 ingot 4 metal 6 pressure 8-11(1). >> BEGIN DATA >> 01 1 3 67.0 >> 02 1 2 71.9 >> 03 1 1 72.2 >> 04 2 3 67.5 >> 05 2 2 68.8 >> 06 2 1 66.4 >> 07 3 3 76.0 >> 08 3 2 82.6 >> 09 3 1 74.5 >> 10 4 3 72.7 >> 11 4 2 78.1 >> 12 4 1 67.3 >> 13 5 3 73.1 >> 14 5 2 74.2 >> 15 5 1 73.2 >> 16 6 3 65.8 >> 17 6 2 70.8 >> 18 6 1 68.7 >> 19 7 3 75.6 >> 20 7 2 84.9 >> 21 7 1 69.0 >> END DATA. >> >> ***** random intercept model *****. >> MIXED pressure BY metal ingot >> /FIXED=metal | SSTYPE(3) >> /METHOD=REML >> /PRINT=CORB COVB DESCRIPTIVES G LMATRIX R SOLUTION TESTCOV >> /RANDOM=INGOT >> /TEST 'metal1 | ingot=1' intercept 1 metal 1 0 0 | ingot 1 0 0 0 0 0 0 >> /TEST 'metal2 | ingot=1' intercept 1 metal 0 1 0 | ingot 1 0 0 0 0 0 0 >> /TEST 'metal3 | ingot=1' intercept 1 metal 0 0 1 | ingot 1 0 0 0 0 0 0 >> /TEST 'metal1 | ingot=2' intercept 1 metal 1 0 0 | ingot 0 1 0 0 0 0 0 >> /TEST 'metal2 | ingot=2' intercept 1 metal 0 1 0 | ingot 0 1 0 0 0 0 0 >> /TEST 'metal3 | ingot=2' intercept 1 metal 0 0 1 | ingot 0 1 0 0 0 0 0 >> /SAVE=FIXPRED PRED. >> >> COMPUTE eblups=PRED_1-FXPRED_1. >> COMPUTE b0 = 71.10000000000004. >> COMPUTE b1_1v3 = -0.9142857142857328. >> COMPUTE b1_2v3 = 4.79999999999998. >> EXECUTE. >> >> IF (ingot=1 and metal=1) yhat_population_avg = b0 + b1_1v3. >> IF (ingot=1 and metal=2) yhat_population_avg = b0 + b1_2v3. >> IF (ingot=1 and metal=3) yhat_population_avg = b0. >> IF (ingot=2 and metal=1) yhat_population_avg = b0 + b1_1v3. >> IF (ingot=2 and metal=2) yhat_population_avg = b0 + b1_2v3. >> IF (ingot=2 and metal=3) yhat_population_avg = b0. >> IF (ingot=3 and metal=1) yhat_population_avg = b0 + b1_1v3. >> IF (ingot=3 and metal=2) yhat_population_avg = b0 + b1_2v3. >> IF (ingot=3 and metal=3) yhat_population_avg = b0. >> IF (ingot=4 and metal=1) yhat_population_avg = b0 + b1_1v3. >> IF (ingot=4 and metal=2) yhat_population_avg = b0 + b1_2v3. >> IF (ingot=4 and metal=3) yhat_population_avg = b0. >> IF (ingot=5 and metal=1) yhat_population_avg = b0 + b1_1v3. >> IF (ingot=5 and metal=2) yhat_population_avg = b0 + b1_2v3. >> IF (ingot=5 and metal=3) yhat_population_avg = b0. >> IF (ingot=6 and metal=1) yhat_population_avg = b0 + b1_1v3. >> IF (ingot=6 and metal=2) yhat_population_avg = b0 + b1_2v3. >> IF (ingot=6 and metal=3) yhat_population_avg = b0. >> IF (ingot=7 and metal=1) yhat_population_avg = b0 + b1_1v3. >> IF (ingot=7 and metal=2) yhat_population_avg = b0 + b1_2v3. >> IF (ingot=7 and metal=3) yhat_population_avg = b0. >> EXECUTE. >> >> IF (ingot=1 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. >> IF (ingot=1 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. >> IF (ingot=1 and metal=3) yhat_subject_specific = b0 + eblups. >> IF (ingot=2 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. >> IF (ingot=2 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. >> IF (ingot=2 and metal=3) yhat_subject_specific = b0 + eblups. >> IF (ingot=3 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. >> IF (ingot=3 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. >> IF (ingot=3 and metal=3) yhat_subject_specific = b0 + eblups. >> IF (ingot=4 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. >> IF (ingot=4 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. >> IF (ingot=4 and metal=3) yhat_subject_specific = b0 + eblups. >> IF (ingot=5 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. >> IF (ingot=5 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. >> IF (ingot=5 and metal=3) yhat_subject_specific = b0 + eblups. >> IF (ingot=6 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. >> IF (ingot=6 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. >> IF (ingot=6 and metal=3) yhat_subject_specific = b0 + eblups. >> IF (ingot=7 and metal=1) yhat_subject_specific = b0 + b1_1v3 + eblups. >> IF (ingot=7 and metal=2) yhat_subject_specific = b0 + b1_2v3 + eblups. >> IF (ingot=7 and metal=3) yhat_subject_specific = b0 + eblups. >> EXECUTE. >> >> On Mon, Oct 3, 2011 at 8:08 PM, R B <[hidden email]> wrote: >>> Carol, >>> >>> There are two types of predicted values that are usually of interest >>> upon fitting a linear mixed model. The first equation with which you >>> are familiar produces population-averaged predicted values. These are >>> calculated using the estimated fixed effects parameters only, i.e., >>> >>> y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk >>> >>> Note: These predicted values will obviously not vary across random >>> subject effects. >>> >>> The second type of predicted values takes into account both the >>> estimated fixed effects parameters and the empirical best linear >>> unbiased predictors ("EBLUPs") of the random effects of each >>> observation, i.e., >>> >>> y_hat = b0_hat + b1_hat*x1 + ... + bk_hat*xk + uj_hat >>> >>> Note: These predicted values can vary across random subject effects. >>> >>> When discussing predicted values we should be clear as to whether >>> we're referring to population-averaged or subject-specific predicted >>> values. >>> >>> It's really difficult for me to determine whether a random effect has >>> been misinterpreted without having the actual article available. >>> >>> Ryan >>> >>> On Mon, Oct 3, 2011 at 1:54 PM, parisec <[hidden email]> wrote: >>>> I'm going to start this as a new thread since my original question >>>> was buried. >>>> >>>> Ryan, >>>> >>>> Best overview of LMM, i've found...thank you. >>>> >>>> My confusion stems from trying to equate the predicted values of a DV >>>> with ordinary regression where in a fixed effects model: >>>> >>>> predY = B0+Bx1+Bx2+error >>>> >>>> As your reference states- For fixed effects: >>>> >>>> If the SES parameter estimate is 2.5, then for a unit increase in SES >>>> the dependent variable (ex., test score) will increase 2.5 units - >>>> the same interpretation as in ordinary regression. So, we're good here. >>>> >>>> >>>> But for interpretation of the random effects: >>>> >>>> Intraclass correlation: interpreting random effects variance >>>> components. Let the variance component estimate for the random factor >>>> id (meaning school id) = 8.61. Let the variance component estimate >>>> for Residual = 39.15. In this null model, since the school variance >>>> component is 18% of the total of both variance components, we would >>>> say that the school effect accounts for 18% of the variance in math scores. >>>> >>>> >>>> So, this would not make sense to plug into the above equation. This >>>> is why i am confused by tables in manuscripts that put the >>>> coefficients for both the fixed and random variables all into one >>>> table and discuss their significance like they all represent coefficients that predict the value of a DV. >>>> >>>> Thanks >>>> Carol >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> >>>> Carol, >>>> >>>> I suggest you take some time reading though the online documentation >>>> to which I have linked below. Write back if you have specific >>>> questions pertaining to fixed versus random effects. >>>> >>>> http://faculty.chass.ncsu.edu/garson/PA765/multilevel.htm >>>> >>>> Ryan >>>> >>>> On Fri, Sep 30, 2011 at 7:29 PM, Parise, Carol A. >>>> <[hidden email]> wrote: >>>>> Hi all, >>>>> >>>>> I have a question on interpretation of the random versus fixed >>>>> effects coefficients that come out of a mixed model. >>>>> >>>>> Fixed effects are somewhat straightfoward. From what i think i >>>>> understand, the coefficient represents a change in the DV for a 1 >>>>> unit change in the variable - just like in a generalized linear model. >>>>> Statistical signficance for an IV is determine by a t-test testing >>>>> whether the coefficient is different from 0. if the CIs of the >>>>> coefficient don't cross zero, you're in business. >>>>> >>>>> For random effects, the output is labeled 'estimate' but the table >>>>> is "covariance parameter estimates" and tells you the percent of >>>>> variance accounted for by the variable of interest. Statistical >>>>> significance is tested using the Wald Z. >>>>> >>>>> My confusion lies when i read the results of publications that use >>>>> linear mixed models. The examples i've been going through tend to >>>>> discuss these factors separately, discussing percent of variance >>>>> accounted for by the random effects and change in DV for increases >>>>> in the IV for the fixed factors. >>>>> >>>>> However, in several publications, both the random and fixed effects >>>>> are included in a single table and state something like "Table 1 >>>>> presents the mixed effects model results estimating the simultaneous >>>>> effects of **random and fixed effects**". They then go on and >>>>> interpret them all the same just as if they were all fixed effects. >>>>> >>>>> I suspect these publications are correct (at least i would hope peer >>>>> review would have caught this type of error) and I am the one who >>>>> is missing something. >>>>> >>>>> Thanks in advance for enlightenment. >>>>> >>>>> Carol >>>>> >>>>> >>>>> >>>>> >>>>> >>>> >>>> ===================== >>>> To manage your subscription to SPSSX-L, send a message to >>>> [hidden email] (not to SPSSX-L), with no body text except >>>> the command. To leave the list, send the command SIGNOFF SPSSX-L For >>>> a list of commands to manage subscriptions, send the command INFO >>>> REFCARD >>>> >>>> >>>> -- >>>> View this message in context: >>>> http://spssx-discussion.1045642.n5.nabble.com/Random-vs-fixed-effects >>>> -LMM-tp4865842p4865842.html Sent from the SPSSX Discussion mailing >>>> list archive at Nabble.com. >>>> >>>> ===================== >>>> To manage your subscription to SPSSX-L, send a message to >>>> [hidden email] (not to SPSSX-L), with no body text except >>>> the command. To leave the list, send the command SIGNOFF SPSSX-L For >>>> a list of commands to manage subscriptions, send the command INFO >>>> REFCARD >>>> >>> >> >> ===================== >> To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD >> > ===================== To manage your subscription to SPSSX-L, send a message to [hidden email] (not to SPSSX-L), with no body text except the command. To leave the list, send the command SIGNOFF SPSSX-L For a list of commands to manage subscriptions, send the command INFO REFCARD |
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