Fixed, Random and Mixed effects Shadi Ghasemi 2/25/2019

Fixed variable Data has been gathered from all the levels of the factor that are of interest. A “fixed variable” is one that is assumed to be measured without error It is also assumed that the values of a fixed variable in one study are the same as the values of the fixed variable in another study. Example: The purpose of an experiment is to compare the effects of three specific dosages of a drug on the response. "Dosage" is the factor; the three specific dosages in the experiment are the levels; there is no intent to say anything about other dosages. Example: age, gender 2/25/2019

Random variable The factor has many possible levels, interest is in all possible levels, but only a random sample of levels is included in the data. is assumed to be measured with measurement error. the values come from and are intended to generalize to a much larger population of possible values with a certain probability distribution (e.g., normal distribution); Example: if collecting data from different medical centers, “center” might be thought of as random Example: if surveying students on different campuses, “campus” may be a random effect Example: speaker, listener 2/25/2019

Random and Fixed Effects The terms “random” and “fixed” are used in the context of ANOVA and regression models, and refer to a certain type of statistical model. Fixed effect: 1: statistical model typically used in regression and ANOVA assuming independent variable is fixed; 2: generalization of the results apply to similar values of independent variable in the population or in other studies; 3: will probably produce smaller standard errors (more powerful). Random effect: 1: different statistical model of regression or ANOVA model which assumes that an independent variable is random; 2: generally used if the levels of the independent variable are thought to be a small subset of the possible values which one wishes to generalize to; 3: will probably produce larger standard errors (less powerful). 2/25/2019

mixed effects A mixed model is a statistical model containing both fixed effects and random effects, that is mixed effects  They are particularly useful in settings where repeated measurements are made on the same statistical units (longitudinal study), or where measurements are made on clusters of related statistical units Because of their advantage to deal with missing values, mixed effects models are often preferred over more traditional approaches such as repeated measures ANOVA. 2/25/2019

introduce ANOVA models appropriate to different experimental objectives 2/25/2019

Model I ANOVA or fixed model 1. Treatment effects are additive and fixed by the researcher 2. The researcher is only interested in these specific treatments and will limit his conclusions to them. 3. Model 𝑌 𝑖𝑗 =𝜇+ 𝜏 𝑖 + 𝜀 𝑖𝑗 where 𝜏 𝑖 will be the same if the experiment is repeated 4. When Ho is false three will be an additional component in the variance between treatments 𝑟 𝜏 𝑖 2 (𝑡−1) 2/25/2019

Model II ANOVA or random model or components of variance model 1. The treatments are a random sample from a larger population of treatments for which the mean is zero and the variance is 𝜎 𝑡 2 2. The objective of the researcher is to extend the conclusions based on the sample of treatments to ALL treatments in the population 3. Here the treatment effects are random variables ( 𝑠 𝑖 ) and knowledge about the particular ones investigated is not important 4. Model 𝑌 𝑖𝑗 =μ+ 𝑠 𝑖 + 𝜀 𝑖𝑗 where 𝑠 𝑖 will be different if the experiment is repeated 2/25/2019

5. When the null hypothesis is false there will be an additional component of variance equal to 𝑟𝜎 𝑠 2 . 6. The researcher wants to test the presence and estimate the magnitude of the added variance component among groups: 𝜎 𝑠 2 . 7. For One Way ANOVA, the computation is the same for the fixed and random Models. However, the objectives and the conclusions are different. The computations following the initial significance test are also different. For factorial ANOVAs the computations are different. 2/25/2019

Differences between fixed and random-effects model 2/25/2019

The objectives are different The sampling procedures are different. The expected sums of the effects are different the expected variances are different 2/25/2019

The objectives are different For the Fixed model: Each level of variable is important. They are not a random sample. The purpose is to compare these specific treatments and test the hypothesis that the treatment effects are the same. For the Random model: The purpose of a random model is to estimate 𝜎 𝑠 2 and 𝜎 𝜀 2 , the variance components ( 𝐻 0 : 𝜎 𝑠 2 =0 𝑣𝑠. 𝐻 1 : 𝜎 𝑠 2 >0) 2/25/2019

The sampling procedures are different For the Fixed model: treatments are not randomized but are selected purposefully by the experiment. If the experiment is repeated, 𝜏 𝑖 ‘s are assumed to be constants and do not change, only 𝜀 𝑖𝑗 's change. For the Random model: treatments are randomly selected and the variance in the population of treatments contributes to the total sum of squares. If the experiment is repeated, not only the errors are changeable but, 𝑠 𝑖 , are changeable. 2/25/2019

The expected sums of the effects are different 2/25/2019

the expected variances are different 2/25/2019

y o u r t o p i c g o e s h e r e 2/25/2019

Two-way classification experiments: fixed, random or mixed 2/25/2019

Consider three different scenarios: Fixed. A breeder is interested in a particular set of varieties in a particular set of locations Random: interested in a random samples of varieties released during different decades in a random set of locations representing a particular region Mixed: interested in a fix varieties released in a random set of locations representing a particular region 2/25/2019

Two-way classification experiments: fixed, random or mixed 2/25/2019

Linear Mixed Models (LMM) 2/25/2019

Generalised LinearMixed Models 2/25/2019

Up until now we have considered models with normally distributed errors. A class of models known as generalised linear models (GLMs) is available for fitting fixed effects models to such non-normal data. These models can be further extended to fit mixed models and are then referred to as generalised linearmixed models (GLMMs). Random effects, random coefficients or covariance patterns can be included in a GLMM in much the same way as in normal mixed models, and again either balanced or unbalanced data can be analysed. 2/25/2019

The GLMM can be defined by As in the GLM, μ is the vector of expected means of the observations and is linked to the model parameters by a link function, g: X and Z are the fixed and random effects design matrices, and α and β are thevectors of fixed and random effects parameters as in the normal mixedmodel. The random effects, β, can again be assumed to follow a normal distribution: 2/25/2019

where R is the residual variance matrix, var(e). V is not as easily specified as it was for normal data where V = ZGZ + R This is because μ is not now a linear function of β. Brown H, Prescott R. Applied Mixed Models in Medicine . Chichester, England: John Wiley 2006. 2/25/2019

Sample size 2/25/2019

THANK YOU t r a n s i t i o n a l p a g e 2/25/2019