12. Repeated Measures Mixed-Modeling (20 Patients)

Mixed models uses repeated outcome measures as well as a predictor variable, often a binary treatment modality. If the main purpose of your research is to demonstrate a significant difference between two treatment modalities rather than between the differences in repeated measures, then mixed models should be used instead of repeated measures analysis of variance (ANOVA). The explanation requires advanced statistics and is given in the next paragraph. It could be skipped by the nonmathematiciens.

With mixed models repeated-measures-within-subjects receive fewer degrees of freedom than they do with the classical general linear model (Chaps. 9, 10 and 11), because they are nested in a separate layer or subspace. In this way better sensitivity is left in the model to demonstrate differences between subjects. Therefore, if the main aim of your research is to demonstrate differences between subjects, then the mixed model should be more sensitive than the classical general linear models as explained in the previous three chapters. However, the two methods should be equivalent, if the main aim of your research is to demonstrate differences between repeated measures, for example different treatment modalities in a single subject. A limitation of the mixed model is, that it includes additional variances, and is, therefore, more complex. More complex statistical models are, ipso facto, more at risk of power loss, particularly, with small data (Statistics applied to clinical studies 5th edition, Chap. 55, Springer Heidelberg Germany 2012, from the same authors). Another limitation is, that the data have to be restructured in order to qualify for the mixed linear analysis.

Is there a significant effect of the predictor after adjustment for the repeated measures.

Twenty patients are treated with two treatment modalities for cholesterol and levels are measured after 1–5 weeks, once a week. We wish to know whether one treatment modality is significantly better than the other after adjustment for the repeated nature of the outcome variables

The entire data file is in “chapter12repeatedmeasuresmixedmodel”, and is in extras.springer.com. We will start by opening the data file in SPSS.

Command:

click Data....click Restructure....mark Restructure selected variables into cases.... click Next....mark One (for example, w1, w2, and w3)....click Next....Name: id (the patient id variable is already provided)....Target Variable: enter "firstweek, secondweek...... fifthweek"....Fixed Variable(s): enter treatment....click Next.... How many index variables do you want to create?....mark One....click Next....click Next again....click Next again....click Finish....Sets from the original data will still be in use…click OK.

Return to the main screen, and observe that there are now 100 rows instead of 20 in the data file. The first 10 rows are given underneath.

The above table is adequate to perform a mixed linear model analysis. For readers’ convenience it is saved in extras.springer.com, and is entitled “chapter12repeatedmeasuresmixedmodels2”. SPSS calls the levels “indexes”, and the outcome values after restructuring “Trans” values, terms pretty confusing to us.

The above table is adequate to perform a multilevel modeling analysis with mixed linear model, and adjusts for the positive correlation between the presumably positive correlation between the weekly measurements in one patient. The module Mixed Models consists of two statistical models:

For analysis the statistical model Linear is required.

The underneath table shows the result. SPSS has applied the effects of the cluster levels and the interaction between cluster levels and treatment modality for adjusting the effects of the correlation levels between the weekly repeated measurements. The adjusted analysis shows that one treatment performs much better than the other.

Sometimes better statistics can be obtained by random effects models. The module Generalized Linear Mixed Models can be used for the purpose.

For a mixed model with random effects the Generalized Mixed Linear Model in the module Mixed Models is required.

In the output sheet a graph is observed with the mean and standard errors of the outcome value displayed with the best fit Gaussian curve. The F-value of 23,722 indicates that one treatment is very significantly better than the other with p <0,0001. The thickness of the lines are a measure for level of significance, and so the significance of the 5 week is very thin and thus very weak. Week 5 is not shown. It is redundant, because it means absence of the other 4 weeks. If you click at the left bottom of the graph panel, a table comes up providing similar information in written form. The effect of the interaction variable is not shown, but implied in the analysis.

The F-value of this random effect model is slightly better than the F-value of the fixed effect mixed model (F = 20,030).

You might want to analyze the above data example in different ways. The averages of the five repeated measures in one patient can be calculated and an unpaired t-test may be used to compare these averages in the two treatment groups (like in Chap. 6). The overall average in group 0 was 1,925 (SEM 0,0025), in group 1 2,227 (SE 0,227). With 18 degrees of freedom and a t-value of 1,99 the difference did not obtain statistical significance, 0,05 < p < 0,10. There seems to be, expectedly, a strong positive correlation between the five repeated measurements in one patient. In order to take account of this strong positive correlation a mixed linear model is used. This model showed that treatment 1 now performed significantly better than did treatment 0, at p = 0,0001.

You might want to analyze the above data file also using a repeated measures ANOVA (like in Chap. 10). However, repeated-measures ANOVA will produce treatment modality effect with a p-value of only 0,048 instead of 0,0001. If you are more interested in the effect of the predictor variables, and less so in the difference between the repeated outcomes, then repeated-measures ANOVA is not an appropriate method for your purpose.

More background, theoretical and mathematical information of restructuring data files is in the Chap.6, Mixed linear models, pp 65–77, in: Machine learning in medicine part one, Springer Heidelberg Germany, 2013, from the same authors, and the Chaps. 8 and 39 in the current volume.