# Chapter 6 Pooled Analysis

We estimated the model coefficients of each completed dataset and conducted model diagnostic based on one completed dataset. We then combined model results of all datasets for inference.

## 6.1 Estimates of model coefficients

In a frequentist setting, estimates of model coefficients are computed through maximum likelihood estimation. In a Bayesian setting, estimates of model coefficients are computed through posterior distribution, which requires the specification of prior distributions.

Prior to seeing the data, we assumed that the intercept ($$\beta_{0}$$) have normal distribution with mean -3 and variance 2. The coefficients of the predictors ($$\beta = \beta_{1}, ..., \beta_{17}$$) have normal distribution with mean 0 and variance 5 - that $$\beta$$ are as likely to be positive as they are to be negative but unlikely to be far away from zero. After observing the data, we modeled the likelihood of each observation as a binomial distribution $$p(y_{i}|x_{i},\beta_{0},\beta_{1},...,\beta_{17}) = Binomial(1, \pi_{i})$$ for $$i = 1,...,n$$; $$\pi_{i}$$ is the probability of getting promoted, where $$\pi_{i} = logit^{-1}(\beta_{0} + \beta \cdot x_{i}) = \dfrac{exp(\beta_{0} + \beta \cdot x_{i})}{1 +exp(\beta_{0} + \beta \cdot x_{i}) }$$. The posterior distribution is proportional to the product of the priors and the likelihood distribution. To sum up, the posterior distribution is derived as below.

Prior Distribution \begin{aligned} f(\beta_{0}) &= N(-3,2)\\ f(\beta_{k}) &= N(0,5), k = 1,...,17 \\ \end{aligned}

Likelihood \begin{aligned} p(Y|x^{cmp},\beta_{0},\beta) & = \prod_{i=1}^{n} p(y_{i}|x_{i},\beta_{0},\beta) \\ & = \prod_{i=1}^{n} [logit^{-1}(\beta_{0} + \beta \cdot x_{i})]^{y_{i}}[1 - logit^{-1}(\beta_{0} + \beta \cdot x_{i})]^{(1-y_{i})}\\ \end{aligned}

Posterior Distribution \begin{aligned} f(\beta_{0},\beta|X,Y) & = f(\beta_{0}){\prod_{k=1}^{17} f(\beta_k)} f(Y|x^{cmp},\beta_{0},\beta) \\ & \propto f(\beta_{0}) {\prod_{k=1}^{17} f(\beta_k)} [\prod_{i=1}^{n} logit^{-1}(\beta_{0} + \beta x_{i})]^{y_{i}}[\prod_{i=1}^{n}(1- logit^{-1}(\beta_{0} + \beta x_{i}))]^{(1- y_{i})} \end{aligned}

Model Diagnostic

Logistic regression does not require the assumptions of constant variance and normality of residuals. To check if the predictors of fitted logistic regression model are well specified, we applied binned residuals technique. The procedures to produce binned residuals include: first, we computed raw residuals of the fitted model and ordered observations by values of predicted probabilities; second, for continuous predictor, we formed $$\sqrt{27346} \approx 165$$ equally sized, ordered bins using the ordered data and computed average residual of each bin. Next, we plotted average residual versus average predicted probability of each bin to check if there are any specific patterns that might cause problems; third, for categorical predictor, we computed average residual of each level.

We created the binned residuals plot (Figure 6.1) based on one imputed dataset’s model (Frequentist model) results. The average residuals are close to 0 for salary less than $82,425, but tend to fluctuate for salary greater than$82,425. That data with higher salary values are more sparse leads to the fluctuation. For categorical variable, the number of level is limited and majority of the time the prediction would yield 0 (not promoted), and thus, the average residuals for each bin are small. In general, we did not observe extreme residuals, but we should be wary of data with higher salary values.

Gender
M F
$$-1.473e^{-10}$$ $$-1.916e^{-10}$$
Occupational Category Residuals
P A O
$$-8.481e^{-11}$$ $$-2.158e^{-10}$$ $$-2.186e^{-10}$$
Education Levels Residuals
B A C D E
$$1.847e^{-10}$$ $$-2.093e^{-10}$$ $$-1.351e^{-10}$$ $$-3.733e^{-11}$$ $$-2.573e^{-11}$$
Race
White Non-white
$$1.570e^{-10}$$ $$-1.869e^{-10}$$
Switch
Switch No-switch
$$-1.549e^{-10}$$ $$-1.658e^{-10}$$
Supervisory Position Residuals
Supervisor Non-supervisor
$$-2.010e^{-10}$$ $$-1.588e^{-10}$$
$$-2.464e^{-10}$$ $$-5.606e^{-12}$$ $$-5.440e^{-12}$$
% Change Residuals
Change2 Change1 Change3
$$-1.888e^{-10}$$ $$-1.184e^{-10}$$ $$-1.351e^{-10}$$

## 6.2 Pooled results

We combined poole results for both Frequentist model and Bayesian model. For Frequentist model, we obtained the pooled results through (1) averaging out the estimated regression coefficients of each completed dataset (2) accounting for both within and across datasets variances. For Bayesian model, we obtained the pooled results through summarizing the mixture draws from posterior distribution.

### 6.2.1 Frequentist model

For each completed dataset $$j$$ ($$j = 1,...,12$$) and each regression coefficient $$\beta_{ij}$$ ($$i = 0,...,17$$), let $$q_{ij}$$ be the estimator of $$\beta_{ij}$$ from $$j_{th}$$ dataset, and let $$u_{ij}$$ be the estimator of the variance of $$\beta_{ij}$$ from $$j_{th}$$ dataset. First, we computed the average of each dataset’s estimated slope to obtain the estimate of the model coefficient $$\overline{q_{i}}$$. Second, we computed the average of each dataset’s estimated slope variance to obtain the within-dataset variance $$\overline{u_{i}}$$. Thid, we computed the variance of the slopes across 12 datasets to obtain the across-datasets variance $$b_{i}$$.

\begin{aligned} & \overline{q_{i}} = \sum_{j=1}^{12}q_{ij}/12 \\ & \overline{u_{i}} = \sum_{j=1}^{12}u_{ij}/12 \\ & b_{i} = \sum_{j=1}^{12}(q_{ij} -\overline{q_{i}})/(12-1) \\ \end{aligned}

With average estimated regression coefficient ($$\overline{q_{i}}$$), within-dataset variance ($$\overline{u_{i}}$$), and across-datasets variance ($$b_{i}$$), the point estimate of $$\beta_{i}$$ is $$\overline{q_{i}}$$, and its variance is $$T_{i} = (1+1/12)b_{i}+\overline{u_{i}}$$. An approximate 95% confidence interval is $$\overline{q_{i}} \pm 1.96\sqrt{T_{i}}$$. The table below shows the pooled estimated slope, standard error, and 95% confidence interval of each regression coefficient.

Combined Coefficient Estimate, Standard Error, and 95% CI. (Frequentist Model)
Coefficient Estimate Std. Error 95% CI.
Intercept ($$\beta_{0}$$) -6.018 0.697 [-6.871, -5.164]
Female ($$\beta_{1}$$) -0.035 0.012 [-0.0815, 0.011]
OccA ($$\beta_{2}$$) 0.744 0.015 [0.709, 0.779]
OccO ($$\beta_{3}$$) 1.005 0.299 [0.845, 1.170]
EducA ($$\beta_{4}$$) -0.712 0.036 [-0.748, -0.675]
EducC ($$\beta_{5}$$) 0.128 0.016 [0.094, 0.161]
EducD ($$\beta_{6}$$) 0.195 0.033 [0.139, 0.252]
EducE ($$\beta_{7}$$) -0.047 0.045 [-0.129, 0.036]
Non-white ($$\beta_{8}$$) -0.112 0.017 [-0.151, -0.074]
No-switch ($$\beta_{9}$$) -0.039 0.031 [-0.057, -0.020]
Non-Supervisor ($$\beta_{10}$$) -0.949 0.013 [-1.008, -0.890]
Grade0 ($$\beta_{11}$$) -0.450 0.066 [-0.657, -0.244]
Grade2 ($$\beta_{12}$$) 1.389 0.049 [1.213,1.565]
ScaledSalary ($$\beta_{13}$$) 1.939 0.618 [1.157, 2.721]
(ScaledSalary + 1.0751) ($$\beta_{14}$$) 2.919 0.642 [2.077, 3.761]
(ScaledSalary - 0.2272) ($$\beta_{15}$$) 3.328 0.902 [2.362, 4.294]
Change1 ($$\beta_{16}$$) -0.266 0.027 [-0.457, -0.074]
Change3 ($$\beta_{17}$$) 0.233 0.025 [0.065, 0.401]

### 6.2.2 Bayesian Model

For Bayesian model, instead of averaging out the estimated regression coefficients, we obtained the pooled results through applying the approach proposed by Zhou and Reiter (2010). For each completed dataset $$l = 1,...,m$$ and parameter of interest $$Q$$, denote $$Q^{(l)}$$ as the MCMC draws from posterior distribution. First, simulate $$J$$ values of posterior draws (where $$J$$ is large) from each $$Q^{(l)}$$, with its distribution denoted as $$\hat{f}(Q^{(l)})$$. Second, mix all $$\hat{f}(Q^{(l)})$$ to form $$\hat{f}(Q^{(all)})$$. Third, sort total number of $$m*J$$ draws from $$\hat{f}(Q^{(all)})$$, where the estimates of 95% posterior interval, posterior median, and other statistics can be obtained from the mixed posterior draws (Zhou & Reiter, 2010).

The pooled estimated slope, standard error, and 95% credible interval are generated through specifying $$J = 4000$$ to create a total number of $$12*4000 = 48000$$ draws. The table below shows the pooled estimated slope, standard error, and 95% credible interval of each regression coefficient.

Combined Coefficient Estimate, Standard Error, and 95% CI. (Frequentist’s Model)
Coefficient Estimate Std. Error 95% CI.
Intercept ($$\beta_{0}$$) -6.033 0.816 [-7.732, -4.529]
Female ($$\beta_{1}$$) -0.036 0.110 [-0.254, 0.176]
OccA ($$\beta_{2}$$) 0.745 0.124 [0.502, 0.988]
OccO ($$\beta_{3}$$) 0.877 0.571 [-0.359, 1.887]
EducA ($$\beta_{4}$$) -0.719 0.191 [-1.103, -0.354]
EducC ($$\beta_{5}$$) 0.127 0.127 [-0.123, 0.375]
EducD ($$\beta_{6}$$) 0.190 0.183 [-0.174, 0.542]
EducE ($$\beta_{7}$$) -0.056 0.212 [-0.479, 0.355]
Non-white ($$\beta_{8}$$) -0.116 0.132 [-0.383, 0.140]
No-switch ($$\beta_{9}$$) -0.029 0.178 [-0.365, 0.326]
Non-Supervisor ($$\beta_{10}$$) -0.950 0.112 [-1.167, -0.729]
Grade0 ($$\beta_{11}$$) -0.458 0.253 [-0.966, 0.025]
Grade2 ($$\beta_{12}$$) 1.395 0.218 [0.969, 1.823]
ScaledSalary ($$\beta_{13}$$) 1.933 0.770 [0.530, 3.559]
(ScaledSalary + 1.0751) ($$\beta_{14}$$) 2.915 0.781 [1.476, 4.538]
(ScaledSalary - 0.2272) ($$\beta_{15}$$) 3.316 0.927 [1.573, 5.205]
Change1 ($$\beta_{16}$$) -0.268 0.159 [-0.586, 0.032]
Change3 ($$\beta_{17}$$) 0.231 0.154 [-0.078, 0.521]

### 6.2.3 Comparison between Frequentist and Bayesian Model

We compared the pooled results for both Frequentist model (Figure 6.2) and Bayesian model (Figure 6.3). The pooled coefficient means of both models are close to each other, but Bayesian model has higher pooled standard errors. There are certain requirements for Bayesian inference after multiple imputation. Our inherent constraints - prior parameter choices and number of available imputed datasets therefore lead to greater uncertainty. First, the posterior distribution depends on the choice of prior parameter, which usually requires knowledge from experts. Our choice of non-informative prior might not be the most ideal prior. Second, Zhou and Reiter (Zhou & Reiter, 2010) suggested that the number of imputed datasets should be large (at least 100 imputed datasets) in order for Bayesian inference to work well. Since there are only 12 available imputed datasets, Bayesian inference might not be valid. Thus, we conducted inference based on the Frequentist model.

## 6.3 Inference based on Frequentist model

In general, the selection of the SES positions is highly dependent on (1) personal skills (indicated by variable Occupational Category and Supervisory Status) (2) education level (indicated by variable Education Group) and (3) performance at work (indicated by variable Salary, Grade, and Change in Pay Rate)

First, for employees whose occupational category are Administrative, the odds of getting promoted are multiplied by a factor of 2.104 (95% CI: 2.031 to 2.180) compared to employees whose occupational category are Professional, holding all else constant. This implies that analytical ability, judgment, discretion, and personal responsibility (characteristics of occupational category Administrative) might be more important than having knowledge in a specific field of science (characteristics of occupational category Professional). Furthermore, for employees who do not serve in supervisory positions, the odds of getting promoted are multiplied by a factor of 0.387 (95% CI: 0.365 to 0.411) than employees who serve in supervisory positions, holding all else constant. Since leadership is one of the key competencies for the SES positions, employees without leadership experiences might be put into disadvantages.

Second, education level also plays an important role when it comes to predicting promotion outcome. For employees with degrees below Bachelor’s, the odds of getting promoted is multiplied by 0.491 (95% CI: 0.473 to 0.509) compared to employees with Bachelor’s degree, holding all else constant. Furthermore, though it is expected that the odds of getting promoted would increase with higher education level, there is certain limit. Specifically, compared to employees with only Bachelor’s degrees, the odds of getting promoted increases if employees acquired Master’s and Advanced degrees. However, the effect of PhD degrees on predicting promotion is not significant because its 95% confidence interval contains 0. Because PhD degrees focus on particular fields of studies, this result further confirms that having specific knowledge in a field of science might not be one of the important promotion criteria.