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Hence, considering only the fixed effect in the presence of between-classroom differences may wrongly lead one to conclude that the effect is negligible, when in fact the effect is positive or stronger in some clusters and negative or weaker in others. To illustrate this, go back to your study and imagine building a simple multilevel logistic regression model. This simple multilevel logistic regression equation is shown below Eq.
In addition to still having two types of parameters pertaining to the intercept the fixed intercept B 00 and the random intercept variance var u 0j , we now have two types of parameters pertaining to the level-1 effect: The fixed slope and the random slope variance.
The fixed slope B 10 is the general effect of the level-1 variable x ij. Once again, in order to interpret B 10 , raise it to the exponent to obtain the odds ratio. Just as for the intercept, this effect may vary from one cluster to another. The residual term associated with the level-1 predictor u 1j will provide information regarding the extent of the effect variation. Specifically, this residual u 1j corresponds to the deviation of the specific effects of the level-1 variable x ij in a given classroom from the overall effect of the level-1 variable x ij across all classrooms the mean of these deviations is assumed to be zero.
The variance component of such a deviation is the random slope variance var u 1j. This is the key element here: The higher the random slope variance, the larger the variation of the effect of GPA from a cluster to another for a graphical representation of the fixed intercept and the random slope variance, see Figure 5.
Note that a non-significant random slope variance would mean that the variation of the effect of GPA is very close to zero and that B 10 is virtually the same in all the classrooms. Graphical representation of the fixed slope B 10 and the residual term associated with the level-1 predictor u 1j cf.
You should have understood that: a multilevel logistic regression enables one to predict the log-odds that an outcome variable equals one instead of zero mark my words: some software packages, e. SPSS, do the opposite and estimate the probability of the outcome being zero instead of one , 4 b the average log-odds is allowed to vary from one cluster to another forming the random intercept variance , and c a lower-level effect may also be allowed to vary from one cluster to another forming the random slope variance.
But where should we begin when running the analysis? The steps of the procedure are as follows:. It should be stressed that this is a simplified version of the procedure usually found in the literature e. The few limitations due to such a simplification are footnoted. Now imagine you have two predictor variables. The first predictor variable is still GPA ranging from 1 to 4. Again, this is a level-1 variable, since it may vary within clusters i.
For the sake of simplicity, we will assume that there is only one teacher per classroom and that teachers are all from the same schools. Note that the independence assumption should be met for level-2 residuals e. The fictitious dataset is provided as supplementary material, in. You will find the syntax files in. In running the syntax file you will obtain the same estimates as those reported in the main text.
In addition, for each software, a series of sub-appendices also provided in supplementary material describes the way to handle each stage of the procedure, namely:. As mentioned in the opening paragraph, SPSS users may not be able complete the procedure as the software often, if not always, fails to estimate the random slope variance in Step 2.
Supplementary material i. First and foremost, you might decide to center the predictor variable s. Although not strictly speaking necessary, such a decision may facilitate the interpretation of some estimates. Centering a predictor variable depends on the level to which it is located. A level-2 predictor variable X j can only be grand-mean centered i. A one-unit increase in the grand-mean centered level-1 variable x1gc ij results in an average change of B 10 in the log-odds that the outcome variable equals one for the overall sample.
In our example, the fixed slope of the grand-mean centered GPA would pertain to the estimation of the general between-pupil effect of GPA, regardless of the classroom. A one-unit increase in the cluster-mean centered level-1 variable x1cc ij results in an average change of B 10 in the log-odds that the outcome variable equals one for a typical cluster.
In our example, the fixed slope of the cluster-mean centered GPA would pertain to the estimation of the within-classroom effect of GPA, comparing the pupils nested in the same classroom the difference between the higher and lower achievers from one class. Beware that the type of centering cluster- vs. The choice of one over the other should be done depending on your specific research question. For instance, grand-mean centering is recommended if you are interested in the effect of a level-2 predictor variable or the absolute between-observation effect of a level-1 predictor variable, whereas cluster-mean centering is recommended when the focus is on the relative within-cluster effect of a level-1 variable.
Note that grand- and cluster-mean centering are applicable to level-1 dichotomous predictors. In such a case, grand-mean centering entails removing the general mean of the dichotomous predictor i. For more detailed information on centering decision, see Enders and Tofighi , as well as some cautionary recommendations on group-mean centering by Kelley, Evans, Lowman and Lykes Now that you have centered your variables, you want to know the extent to which the odds that the outcome variable equals one instead of zero varies from one cluster to another.
In our example, you want to estimate the proportion of variability in the chance of owning an album rather than not owning it that lies between classrooms. The ICC quantifies the degree of homogeneity of the outcome within clusters. The ICC represents the proportion of the between-cluster variation var u 0j in your case: the between-classroom variation of the chances of owning the album in the total variation in your case: the between- plus the within-classroom variation of the chances of owning an album.
The ICC may range from 0 to 1. When the ICC is not different from zero or negligible, one could consider running traditional one-level regression analysis. For more detailed information on intraclass correlation coefficient in multilevel logistic regression, see Wu, Crespi, and Wong Now that you know the extent to which the odds vary from one cluster to another, you want to know the extent to which the effect of the relevant lower-level variable s varies from one cluster to another.
There is a debate in the literature, with some authors advocating the use of maximal model estimating all random slope variance parameters Barr et al. Our position is that random variations should primarily be tested when having theoretical reasons to do so. In our example, you surely want to estimate the variation of the effect of GPA on the odds of owning the album from one classroom to another, since you expect the effect of GPA to depend on some teacher characteristics.
Note that the constrained intermediate model does not contain cross-level interactions, since the model precisely aims to estimate the unexplained variation of lower-level effects. The augmented intermediate model is similar to the constrained intermediate model, with the exception that it includes the residual term associated with the relevant level-1 variable, thereby estimating the random slope variance if you have several relevant lower-level variables, test them one at a time; for the procedure see the Notes of the relevant Sub-Appendix C.
Note that only main level-1 terms are thought to vary, not interaction terms. No need to not look at the coefficient estimates or variance components of the intermediate models. Your goal is to determine whether the augmented intermediate model achieves a better fit to the data than the constrained intermediate model. In other words, your goal is to determine whether considering the cluster-based variation of the effect of the lower-level variable improves the model.
Below is the formula of the likelihood-ratio test Eq. The deviance is a quality-of- mis fit index: The smaller the deviance, the better the fit. Now that you know the extent to which the effect of the relevant lower-level variable varies from one cluster to another and have decided whether to consider the variation of the level-1 effect and keep estimating the random slope variance in the final model or not , you can finally test your hypotheses.
The final model equation is shown below Eq. What about the children? It is now time to take a look at the odds ratios and to discover how pupils behave and whether the data support your hypotheses. Interpretation of the main effect. Congruent with your teacher-to-pupil socialization hypothesis, this indicates that pupils whose classroom teacher is a belieber have 7.
For more detailed information on confidence intervals, see Cumming, Interpretation of the interaction effect. Regarding your interaction hypothesis, this is a bit more complicated. In multilevel logistic regression, the coefficient estimate of the product term does not correspond mathematically to the interaction effect.
Technically, your software calculates the coefficient estimate of the product term as for any main effect i. In logistic regression, the sign, the value, and the significance of the product term is likely to be biased, which has made some authors advocate calculating the correct interaction effect using special statistical package e.
However, the calculation of the correct interaction effect or the correct cross-partial derivative is quite complex and there is no statistical package available for multilevel modeling. Pending better approach, scholars might rely on the simple significance-of-the-product-term approach. This is what we do here. To be interpreted, the interaction needs to be decomposed: We want to know the effect of the level-2 predictor variable for each category of the level-1 variable this could have been vice versa.
Decomposing the interaction may be done using two dummy-coding models e. In both models, the random slope component will have to remain the same. In other words, the residual term associated with the level-1 predictor u 1j will have to remain centered.
This is due to that fact that there now are fewer unexplained variations of the effect of GPA from one classroom to another, since teacher fondness for Bieber accounts for part of these variations. If you observe such a phenomenon, it is not necessarily an issue. Finally, using one of the syntax files provided with the article, you can compare the coefficient estimates obtained in the final model, with or without the use of multilevel modelling.
You will realize that standard errors are deflated when using the traditional one-level logistic regression, thereby increasing the risk of Type I error. Know that multilevel logistic regression may be applied to other types of research designs, data structures, or outcome variables. In such a situation, observations are nested in participants e. Our three-step procedure may well be used in this case with participant number as the level-2 identifier , although the database will have to be rearranged in the preliminary phase in order to have one line per lower-level units for the Stata, R, and SPSS commands, see the relevant Sub-Appendix E; the current version of Mplus does not perform data reshaping.
In a two-level cross-classified data structure, pupils level 1 could for example be nested in two non-hierarchical clusters: the school they attend level 2a and the neighborhood they live in level 2b; see Goldstein, Our three-step procedure is incomplete in this case, as two ICCs would have to be calculated in Step 1 there is level-2a and a level-2b random intercept variance and various random slope variance could be estimated in Step 2 for a given level-1 variable, there are level-2a and level-2b random slopes variance; for the Stata, R, and Mplus commands, see the relevant Sub-Appendix F; SPSS commands are not given due to software limitation.
Count outcome variables typically correspond to a number of occurrences e. Our three-step procedure is to be modified in this case, as multilevel Poisson regression or negative binomial multilevel regression have to be carried out. Note that these regression models give incidence rate ratio rather odds ratio for the Stata, R, and Mplus commands, see the relevant Sub-Appendix G; SPSS commands are not given due to software limitation. Reading this article, you have understood that logistic regression enables the estimation of odds ratio and confidence interval, describing the strength and the significance of the relationship between a variable and the odds that an outcome variable equals one instead of zero.
Moreover, now you know that multilevel logistic regression enables to estimate the fixed intercept and random intercept variance i. Life is worth living, so live another day. Just for fun. It should not be understood in terms of mathematical randomness. Although the covariance structure is usually tested in multilevel modeling procedures, the results are rarely interpreted Hox, In the same way as for the random slope variance, we argue that this covariation should be primarily tested when having theoretical reasons to do so.
To determine whether including the covariance parameter improves the model, one should include it in the augmented intermediate model. In this situation, the augmented intermediate model estimates two more terms than the constrained intermediate model i.
Hence, the likelihood-ratio test will have two degrees of freedom instead of one. None of the authors is actually a fan of Justin Bieber. Aguinis, H. Best-practice recommendations for estimating cross-level interaction effects using multilevel modeling. Ai, C. Interaction terms in logit and probit models. Baayen, R. Mixed-effects modeling with crossed random effects for subjects and items. Barr, D. Random effects structure for confirmatory hypothesis testing: Keep it maximal. Bates, D.
Parsimonious mixed models. Bressoux, P. Cumming, G. The new statistics: Why and how. Cutrona, C. Neighborhood context, personality, and stressful life events as predictors of depression among African American women. Enders, C. Centering predictor variables in cross-sectional multilevel models: a new look at an old issue.
Felps, W. Gelman, A. Goldstein, H. Multilevel Statistical Models. London, UK: Arnold. Greene, W. Testing hypotheses about interaction terms in non-linear models. Heck, R. Hosmer, D. Applied Logistic Regression. Hox, J. Multilevel Analysis: Techniques and Applications. Hove, UK: Routledge. Applied Multilevel Analysis. Amsterdam, Netherland: TT-publikaties. Jaeger, T. Judd, C. Abingdon, UK: Routledge. Bruxelles, Belgium: De Boeck. Treating stimuli as a random factor in social psychology: a new and comprehensive solution to a pervasive but largely ignored problem.
Karaca-Mandic, P. Interaction terms in nonlinear models. Kelley, J. Group-mean-centering independent variables in multi-level models is dangerous. Kenny, D. Dyadic Data Analysis. King, G. Statistical models for political science event counts: Bias in conventional procedures and evidence for the exponential Poisson regression model.
Kolasinski, A. Seattle, WA: University of Washington. LaHuis, D. Explained variance measures for multilevel models. Maas, C. Sufficient sample sizes for multilevel modeling. Thanks a lot for help. Tags: None. Joseph Coveney. At Stata's command line, type Code:. Comment Post Cancel. David Osey.
Yousufzai - the simplest way to do this, is to open the spss data sets and go to file and save as stata file. This will transfer the spss data to a stata data whiles still keeping a version of the spss one too. Marcos Almeida.
You may save. Although the initial poster gives way too few details, I would guess that access to SPSS is not an option. Otherwise the question would be better posted in an SPSS forum. So Joseph's advice is probably the best shot here. Best Daniel. Thank you all for the help, problem solved now. Bruce Weaver. Even though Yousufzai has solved the immediate problem, I'll add this link for future reference. This is just to say that free download and much more of.
For those who do not have access to SPSS, they can easily open the. SAV or. SYS or. POR to. ZSAV file format. Doug Hemken. By far.