Chapter 17 Random Effects and Mixed Effects Models Example. Solutions of alcohol are used for calibrating Breathalyzers. The following data show the alcohol concentrations of samples of alcohol solutions taken from six bottles of alcohol solution randomly selected from a large batch. The objective is to determine if all bottles in the batch have the same alcohol concentrations. Bottle Concentration 1 1.4357 1.4348 1.4336 1.4309 2 1.4244 1.4232 1.4213 1.4256 3 1.4153 1.4137 1.4176 1.4164 4 1.4331 1.4325 1.4312 1.4297 5 1.4252 1.4261 1.4293 1.4272 6 1.4179 1.4217 1.4191 1.4204 We are not interested in any difference between the six bottles used in the experiment. We therefore treat the six bottles as a random sample from the population and use a random effects model. If we were interested in the six bottles, we would use a fixed effects model. 17.3. One Random Factor 17.3.1. The Random‐Effects One‐way Model For a completely randomized design, with v treatments of a factor T, the one‐way random effects model is , :0, , :0, , all random variables are independent, i=1, ..., v, t=1, ..., . Note now , and are called variance components. Our interest is to investigate the variation of treatment effects. In a fixed effect model, we can test whether treatment effects are equal. In a random effects model, instead of testing : ..., against : treatment effects not all equal, we use : 0,: 0. If =0, then all random effects are 0. If 0, random effects vary and can not be all zero. 17.3.2. Analysis of Variance As in the one‐way fixed effects model, the decomposition holds SStotal=SST+SSE, where SST and SSE are calculated exactly the same as in the fixed effects one‐way model: , However, the expected value (i.e., the mean) of MST are different as in the fixed effects model: Source Degrees of freedom Expected Mean Square F test (EMS) Treatment 1 ( equals if MST/MSE each treatment replicates times Error Total 1 If 0 , then E(MST)=E(MSE)= and F=MST/MSE still has an , distribution. We will reject if F ,, . The unbiased estimators: For : MSE. For : (MST‐MSE)/c 8.2 17.6. Two or More Random Effects Suppose both factors A and B are random factors, the two way complete model with random effects is , :0, , :0, , :0, , all are independent i=1, ..., a, b=1, ..., b, t=1, ..., r, (equal sample sizes) As in a two‐way complete model with fixed effects, , The calculation of each sum of sequares is the same as in two‐way fixed effects model. Source DF SS MS EMS F‐ratio A 1 B 1 AB 1 1 Error 1 A Rule for Determining an Expected MS: First note the subscripts on the term representing the factor in the model. Write down a variance component for the factor of interest, for the error, and for every interaction involving the factor. Multiply each variance components bye th corresponding number of observations. Add up terms. Hypothesis Testing: : 0 (No interaction) against : 0. Reject if . ,, : 0 against : 0. Reject the null hypothesis if . ,, Note the denominator is not ! : 0 against : 0. Reject the null hypothesis if . ,, .