SMHS FixedRandomMixedModels

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Scientific Methods for Health Sciences - Fixed, Randomized and Mixed Effect Models

Overview:

Fixed model is a statistical model that represents the observed quantities in terms of explanatory variables treated as if they are non-random, while random model assumes that the dataset being analyzed consist of a hierarchy of different population whose differences relate to that hierarchy and mixed model consists of both fixed effects and random effects. For random effects model and mixed models, either all or part of the explanatory variables are treated as if they rise from random causes. In this section, we are presenting the theory and practice of these three types of models.

Motivation:

We should pay attention to the types of variables we are going to include in the model in order to choose the right model to fit our data. Consider the case where all the explanatory variables are treated as non-random, a fixed-effect model would be the choice, similarly, if the explanatory variables are considered to arise from random causes. Of course, mixed model enable us to deal with both random and fixed effects as a mixture in the study. The question would be how do these model work and differentiate from each other? How do we implement those models in our study?

Theory

1) Fixed effects model: a statistical model that represents the observed quantities in terms of explanatory variables treated as if they are non-random. The fixed effects estimator is used to refer to an estimator for the coefficients in the regression model and we impose time independent effects for each entity when we assume fixed effects for our model.

  • The fixed effect model can be used to control for unobserved heterogeneity when this heterogeneity is constant over time and correlated with independent variables. The constant can be removed by differencing.
  • The fixed effect assumption: the individual specific effect is correlated with the independent variables. This is the priority criteria in using the fixed effect model.
  • The fixed effect model: consider the linear unobserved effects model for $N$ observations and $T$ time periods: $y_{it}=X_{it}\beta + \alpha_{i} + \mu_{it}, for t = 1, 2, \cdots, T$ and $i = 1, 2, \cdots, N$, where $y_{it}$ is the dependent variable observed for individual $i$ and time $t$, $X_{it}$ is the time variant $1xk$ regressor matrix, $\alpha_{i}$ is the unobserved time invariant individual effect and $u_{it}$ is the error term. $\alpha_{i}$ can’t be observed by the econometrician, common examples for time invariant effects $\alpha_{i}$ are innate ability for individuals or institutional factors for countries. The model allows $\alpha_{i}$ to be correlated with the regressor matrix, $x_{it}.$
  • $\alpha_{i}$ is not observable and cannot be directly controlled for. The fixed effect model eliminates $\alpha_{i}$ by demeaning the variables using the within transformation: $y_{it} - \bar{y_{i}} = (X_{it} - \bar{X_{i}})\beta + (\alpha_{i} - \bar{\alpha_{i}}) + (\mu_{it} - \bar{\mu_{it}}) \Rightarrow \ddot{y_{it}}=\ddot{X_{it}}\beta+\ddot{\mu_{it}}$, where $\bar{X_{i}}=\frac{1}{T}\sum_{t=1}^{T}X_{it}$, and $\bar{\mu_{i}}=\frac{1}{T} \sum_{t=1}^{T}\mu_{it}$, since $\alpha_{i}$ is constant, $\bar{\alpha_{i}}=\alpha_{i}$, hence the effect is eliminated. The fixed effect estimator $\hat{\beta_{FE}}$ is then obtained by an OLS regression of $\ddot{y}$ on $\ddot{X}.$
  • Equality of Fixed Effects (FE) and First Differences (FD) estimator with the special case of T=2. That is the FE estimator effectively doubles the data set used in the FD estimator. $FE_{T=2}=[(x_{i1}-\bar{x_{i}})(x_{i1}-\bar{x_{i}})'+(x_{i2}-\bar{x_{i}})(x_{i2}-\bar{x_{i}})']^{-1}[(x_{i1}-\bar{x_{i}})(y_{i1}-\bar{y_{i}})+(x_{i2}-\bar{x_{i}})(y_{i2}-\bar{y_{i}})],$ since each $(x_{i1}-\bar{x_{i}})$ can be rewritten as $\frac{x_{i1}-x_{i2}}{2},$

so $FE_{T=2}=[\sum_{i=1}^{N}\frac{x_{i1}-x_{i2}}{2}\frac{x_{i1}-x_{i2}'}{2}+\frac{x_{i2}-x_{i1}}{2}\frac{x_{i2}-x_{i1}'}{2}]^{-1}[\sum_{i=1}^{N}\frac{x_{i1}-x_{i2}}{2}\frac{y_{i1}-y_{i2}}{2}+\frac{x_{i2}-x_{i1}}{2}\frac{y_{i2}-y_{i1}}{2}]$

$=[\sum_{i=1}^{N}2\frac{x_{i1}-x_{i2}}{2}\frac{x_{i2}-x_{i1}'}{2}]^{-1}[\sum_{i=1}^{N}2\frac{x_{i2}-x_{i1}}{2}\frac{y_{i2}-y_{i1}}{2}]$

$=[\sum_{i=1}^{N}(x_{i2}-x_{i1}x_{i2}-x_{i1}']^{-1}\sum_{i=1}^{N}(x_{i2}-x_{i1})(y_{i2}-y_{i1})=FD_{T=2}.$

  • Hausman-Taylor method: need more than one time-variant regressor $(X)$ and time invariant regressor $(Z)$ and at least one $X$ and one $Z$ that are uncorrelated with $\alpha_{i}$, partition the $X$ and $Z$ variables such that $X=[X_{1it}\vdots X_{2it}], Z=[Z_{1it}\vdots Z_{2it}]$ where $X_{1}$ and $Z_{1}$ are uncorrelated with $\alpha_{i}.$ Need $K1 > G2.$ Estimating $\gamma$ via OLS on $\hat{di} = Z_{i}\gamma + \varphi_{it},$ using $X_{1}$ and $Z_{1}$ as instruments yields a consistent estimate.
  • Testing FE vs. RE with a Hausman test: $H_{0}: \alpha_{i}$ independent from $X_{it}$, $Z_{it}$ vs. $H_{a}: \alpha_{i}$ not independent $X_{it}$, $Z_{it}$. If $H_{0}$ is true, both $\hat{\beta_{RE}}$ and $\hat{\beta_{FE}}$ are consistent, but only $\hat{\beta_{RE}}$ is efficient. If $H_{a}$ is true, $\hat{\beta_{FE}}$ is consistent and $\hat{\beta_{RE}}$ not. $\hat{Q}$ = $\hat{\beta_{RE}}-\hat{\beta_{FE}}$, $\hat{HT}= T\hat{Q’}[Var(\hat{\beta_{FE}})-var(\hat{\beta_{RE}})]^{-1}\hat{Q} \sim \chi_{k}^2,$ where $K = dim(Q)$. The Hausman test is a specification test so a large test statistic might be indication that there might be errors in variables or the model is misspecified. If the fixed effect assumption is true, $\hat{\beta_{LD}} \approx \hat{\beta_{FD}} \approx \hat{\beta_{FE}}.$
  • Steps for fixed effects model:
    • Calculate group and grand means,
    • Calculate $k$=number of groups, $n$=number of observations per group, $N$=total number of observations $(k * n)$,
    • Calculate SS-total (or total variance) as: $(\text{ Each } \text{ score } - \text{ Grand } \text{ mean })^2$ then summed,
    • Calculate SS-treat (or treatment effect) as: $(\text{ Each } \text{ group } \text{ mean }- \text{ Grand } \text{ mean })^2$ then summed $x + n$,
    • Calculate SS-error (or error effect) as $(\text { Each } \text{ score } - \text { Its } \text{ group }\text{ mean })^2$ then summed,
    • Calculate df-total: N-1, df-treat: k-1 and df-error k(n-1),
    • Calculate Mean Square MS-treat: SS-treat/df-treat, then MS-error: SS-error/df-error,
    • Calculate obtained f value: MS-treat/MS-error,
    • Use F-table or probability function, to look up critical f value with a certain significance level,
    • Conclude as to whether treatment effect significantly affects the variable of interest.

2) Random effects model:a statistical model in which the dataset being analyzed consist of a hierarchy of different population whose differences relate to that hierarchy. It is in contrast of the fixed effects model where the data being studied on consist only of non-random variables.

  • The random effects model can be applied to control for unobserved heterogeneity when this heterogeneity is constant over time and correlated with independent variables. The constant can be removed from the data through differencing.
  • The random effect assumption: individual specific effects are uncorrelated with the independent variables. The random effects model is efficient and consistent only when this assumption is met.
  • Suppose $m$ schools are chosen randomly from the united states to study on the math score of the 7th grade students. Suppose that $n$ students of the same age are randomly chosen from each selected school and their math scores are recorded. Let $Y_{ij}$ be the score of the $j^{th}$ student at the $i^{th}$ school. A simple model can be fitted $Y_{ij} = \mu + \mu_{i} + W_{ij}$, where $\mu$ is the average test score for the entire population. In this model, $\mu_{i}$ is the school-specific random effect, which measures the difference between the average math score at school $i$ and the average math score in the entire country. It is considered to be random because the school are randomly chosen form the contry. $W_{ij}$ is the individual-specific error. This is also random since the students within the school are random chosen. We can augment the model further by adding in additional explanatory variables, say teacher to student ratio.
  • Variance components: the variance of $Y_{ij}$ is the sum of the variance $\tau^2$ and $\sigma^2$ of $U_{i}$ and $W_{ij}$ respectively. Let $\bar{Y_{i\cdot}}=\frac{1}{n} \sum_{j=1}^{n} Y_{ij}$ be the average of the $i^{th}$ school that are included in the random sample. Let $\bar{Y_{\cdot\cdot}} = \frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}{n}Y_{ij}$ be the grand average. Let $SSW = \sum_{i=1}^{m}\sum_{j=1}^{n}(Y_{ij}-\bar{Y_{i\cdot}})^2, SSB = n\sum_{i=1}^{m}(\bar{Y_{i\cdot}}-\bar{Y_{\cdot\cdot}})^2 $ are the sum of square due to differences within groups and the sum of squares due to difference between groups respectively. $\frac{1}{m(n-1)}E(SSW)=\sigma^2, \frac{1}{(m-1)n}E(SSB)=\frac{\sigma^2}{n+\tau^2}.$ This can be used to estimate the variance components $\sigma^2$ and $\tau^2$.
  • Comments on random effects model: in general the random effects model are efficient and should be used if the assumptions underlying are satisfied. For random effect to work in the school examples, the school-specific effects should be uncorrelated to other covariates of the model. A Hausman specification test can be used to test on this assumption as described in the previous Fixed Effects Model.

3) Mixed models: a statistical model containing both fixed and random effects, which is widely used in varieties of disciplines, particularly useful for situations where repeated measurements are made on the same statistical units (longitudinal study), or where measurements are made on clusters of related statistical units.

  • Mixed model formula (in matrix form): $y = X\beta + Zu+ \epsilon$, where $y$ is a vector of observations with mean $E(y) = X\beta$, $\beta$ is a vector of fixed effects, $\mu$ is a vector of random effects with mean $E(\mu)=0$ and variance-covariance matrix $var(\mu)=G$, $\epsilon$ is a vector of i.i.d. random error terms with mean $E(\epsilon)=0$ and variance $var(\epsilon)=R,$ $X$ and $Z$ are matrices of regressors relating the observations $y$ to $\beta$ and $\mu$, respectively.
  • Estimation of Henderson’s mixed model equations (MME): $\begin{pmatrix} X'R^{-1}Z&X'R^{-1}Z\\ Z'R^{-1}X&Z'R^{-1}Z+G^{-1} \end{pmatrix} \bigl(\begin{smallmatrix} X'R^{-1}y\\Z'R^{-1}y \end{smallmatrix}\bigr).$ The solutions to the MME, $\tilde{\beta}$ and $\tilde{ \mu}$ are the best linear unbiased estimates (BLUE) and predictors for $\beta$ and $\mu$, respectively. When the conditional variance is known, the inverse variance weighed least squares estimate is BLUE, however, the conditional variance is rarely known. So it is desirable to jointly estimate the variance and weighted parameter estimates when solving MMEs. One method to fit such mixed models is that of EM algorithm, which can be implemented by lme in the nlme library in the R package.

4) Comparing fixed effects model and random effects model:

  • Estimating the summary effect: under the fixed effect model, it is assumed that the true effect size for all studies is identical and the only reason the effect size varies between studies is sampling error. In situations like this, assigning equal weights to different studies largely ignore the information in smaller studies since more information are known about the same effect size in larger studies. In contrast, the random effect model aims to estimate the mean of a distribution of effects. The estimate provided by that study may be imprecise, but it is information about an effect that no other study has estimated. We cannot discount a small study by giving it a small weight, not can we give too much weight to a large study. Hence, in fixed-effect model, there is a wide range of weights while the weights in the random effects model fall in a relatively narrow range. Study weights are more balanced under the random effect model than the fixed effect model.
  • A fixed effect meta analysis estimates a single effect that is assumed to be common to every study while a random effects meta analysis estimates the mean of a distribution of effects.
  • Moving from fixed effect to random effects, extreme studies will lose influence if they are large and will gain influence if they are small.
  • Confidence interval: for the fixed effect model, the only source of uncertainty is the within study error while the random effects model has an additional source of between studies variance. The confidence intervals for the summary effect are wider under the random effect model than under the fixed effect model.
  • The hypothesis tested: with the fixed effect model, the null hypothesis tested is that there is zero effect in every study while that in the random effect model is that the mean effect is zero.
  • The selection of the right model is based on our expectation about whether or not the study shares a common effect size and on our goals in performing the analysis. We use fixed effect model if the following two conditions are satisfied: (1) all the studies included in the analysis are functionally identical; (2) the goal to compute the common effect size for the identified population and not to generalized to other population. By contrast, when we are accumulating data from a series of studies performed operating independently, it would be unlikely that all the studies are functionally equivalent. They differ in ways that would impact the result, and we cannot assume a common effect size. The random effect model would be easily justified than fixed effect model for cases like this. The selection of a model must be based solely on the question of which model fits the distribution of effect sizes, and takes account of the relevant sources of error.
  • For the fixed effect model, we assume that all dispersion in observed effects is due to sampling error but with random effects model we allow some of that dispersion reflects real difference in effect size across studies.

Applications

1) This article proposed approximations for standard errors of estimators of fixed, and random effects in mixed linear models. Best linear unbiased estimators of the fixed and random effects of mixed linear models are available when the true values of the variance ratios are known. If the true values are replaced by estimated values, the mean squared errors of the estimators of the fixed and random effects increase in size. The magnitude of this increase is investigated, and a general approximation is proposed. The performance of this approximation is investigated in the context of (a) the estimation of the effects of the balanced one-way random model and (b) the estimation of treatment contrasts for balanced incomplete block designs.

2) This article discussed about the fixed, mixed and random models. Some explicit questions are raised regarding the adequacy of assumed linear models as a basis for the interpretation of the analysis of variance of randomized experiments. A generally applicable method for the derivation of a linear statistical model, based on the experimental situation and the design of the experiment, is exemplified. The central features of the method are the notion of “experimental unit,” the concept of “true response,” and the use of randomization in the design. A model is derived for the case where two factors having A and B levels respectively, are to be examined with respect to a population of P experimental units, where selection of levels of the factors to be tested, selection of experimental units to be used, and the allocation of selected treatment combinations to units is at random. First a linear population model is given, based on the structure of the experimental situation, whose components are (unknown) parameters of the population of (conceptual) “true” responses. Then the conditions of the design are imposed to obtain a linear statistical model whose components involve the parameters of the population model and some defined random variables reflecting the experimental procedure and design. The derived statistical model is then used to obtain expected mean squares in the analyses of variance. A second illustration of the general methodology is given for a more complex example originated by Vaurio and Daniel and discussed statistically by Scheffé. Some differences from Scheffé's results are noted.

Software

[1] lme (linear mixed effects) function in the nlme library

RCODE:
library(nlme)
library(lattice) ## for Trellis graphics
data(MathAchieve)
MathAchieve[1:10,]   # first 10 students; relates to 7185 students; SES:the socioeconomic status
of the student’s family, centered to the overall mean of 0; MathAch: the student’s score on a 
math-achievement test; Sector: factor coded ‘Catholic’or ‘Public’; MEANSES: mean SES for student 
in each school.
Grouped Data: MathAch ~ SES | School
   School Minority    Sex    SES MathAch MEANSES
1    1224       No Female -1.528   5.876  -0.428
2    1224       No Female -0.588  19.708  -0.428
3    1224       No   Male -0.528  20.349  -0.428
4    1224       No   Male -0.668   8.781  -0.428
5    1224       No   Male -0.158  17.898  -0.428
6    1224       No   Male  0.022   4.583  -0.428
7    1224       No Female -0.618  -2.832  -0.428
8    1224       No   Male -0.998   0.523  -0.428
9    1224       No Female -0.888   1.527  -0.428
10   1224       No   Male -0.458  21.521  -0.428
data(MathAchSchool)
MathAchSchool[1:10,]  # fist 10 schools; relates to 160 schools
    School Size   Sector PRACAD DISCLIM HIMINTY MEANSES
1224   1224  842   Public   0.35   1.597       0  -0.428
1288   1288 1855   Public   0.27   0.174       0   0.128
1296   1296 1719   Public   0.32  -0.137       1  -0.420
1308   1308  716 Catholic   0.96  -0.622       0   0.534
1317   1317  455 Catholic   0.95  -1.694       1   0.351
1358   1358 1430   Public   0.25   1.535       0  -0.014
1374   1374 2400   Public   0.50   2.016       0  -0.007
1433   1433  899 Catholic   0.96  -0.321       0   0.718
1436   1436  185 Catholic   1.00  -1.141       0   0.569
1461   1461 1672   Public   0.78   2.096       0   0.683
attach(MathAchieve)
attach(MathAchSchool)
mses <- tapply(SES, School, mean)  ## school means
mses[as.character(MathAchSchool$School[1:10])]   # for first 10 schools


Check the R example in the attached file at http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-mixed-models.pdf .

Problems

I like the above example – but can we use one of the SOCR Datasets SOCR Data to demonstrate the same linear mixed effect modeling in a different setting?

For example, can we replicate the same analysis protocol using either Ozone Data (where location acts as a grouping variable, school in the previous example),

PRB Data or

World Countries Rankingsdatasets?

References

Statistical inference / George Casella, Roger L. Berger

Sampling / Steven K. Thompson

Sampling theory and methods / S. Sampath





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