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From t-test to multilevel analyses Del-1 Stein Atle Lie, statistician, professor Uni Health, Uni Research http://folk.uib.no/msesl/GLM Outline Pared t-test (Mean and standard deviation) Two-group t-test (Mean and standard deviations) Linear regression GLM (general linear models) GLMM (general linear mixed model) … PASW (former SPSS), Stata, R, gllamm (Stata) Multilevel models “Same thing – many names”: Random effects models Mixed effects models Variance component models Frailty models (in survival analyses) Latent variables Objective Take the general thinking from simple statistical methods into more sophisticated datastructures and statistical analyses Focus on the interpretation of the results with respect to those found in basic statistical methods Multilevel data Types of data: Repeated measures for the same individual The same measure is repeated several times on the same individual Several observers have measured the same individual Several different measures for the same individual A categorical variable with ”many” levels (multicenter data, hospitals, surgeons,…) Null hypotheses In ordinary statistics (using both pared and two-sample t-tests) we define a null hypothesis. H0: m1 = m2 We assume that mean from group (or measure) 1 is equal to the mean from group (or measure) 2. Alternatively H0: D = m1-m2 = 0 p-value Definition: “If our null-hypothesis is true - what is the probability to observe the data* that we did?” * And hence the mean, t-statistic, etc… p-value We assume that our null-hypothesis is true (m0=0 or m1-m2=0) We observe our data Mean value etc. Under the assumption of normal distributed data p-value The p-value is the probability to observe our data (or something more extreme) under the given assumptions -2 0 X 2 m0 4 6 8 X Pared t-test The straightforward way to analyze two repeated measures is a pared t-test. Measure at time1 or location1 (e.g. Data1) is directly compared to measure at time2 or location2 (e.g. Data2) Is the difference between Data1 and Data2 (Diff = Data1-Data2) unlike 0? Pared t-test (n=10) PASW: T-TEST PAIRS=Data1 WITH Data2 (PAIRED). Pared t-test The pared t-test will only be performed for complete (balanced) data. What happens if we delete two observations from data2? (Only 8 complete pairs remain) Pared t-test (n=8) PASW: T-TEST PAIRS=Data1 WITH Data2 (PAIRED). Excel Two group t-test If we now consider the data from time1 and time2 (or location1 and location2) to be independent (even if their not) and use a two group t-test on the full dataset, 2*10 observations Two group t-test (n=20 [10+10]) PASW: T-TEST GROUPS=Grp(1 2) /VARIABLES=Data. Two group t-test Observe that mean for Grp1 and Grp2 is equal to mean for Data1 and Data2 And that the mean difference is also equal The difference between pared t-test and two group t-test lies in the Variance - and the number of observations and therefore in the standard deviation and standard error and hence in the p-value and confidence intervals Two group t-test The two group t-test are performed on all available data. What happens if we delete two observations from Grp2? (Only 8 complete pairs remain - but 18 observations remain) Two group t-test (n=18 [10+8]) PASW: T-TEST GROUPS=Grp(1 2) /VARIABLES=Data.