Generalised Linear Models 1

Sep 9, 2023 14 min read

Generalized linear models

Exponential Family The theory of GLMs is developed for data with distribution given y the exponential family. The form of the data distribution that is useful for GLMs is

$$ f(y;\theta, \phi) = \exp(\frac{\theta y - b(\theta)}{a(\phi)} + c(y, \phi)) $$

where

$\theta$ is called the natural parameter
$\phi$ is called the dispersion parameter

Note:

This family includes the [Gamma], [Normal], [Poisson], and other. For all parameterization of the exponential family, check this

Example

if we have $Y \sim N(\mu, \sigma^2)$

$$ \begin{aligned} f(y; \mu, \sigma^2) &= \frac{1}{(2\pi \sigma^2)^{1/2}}\exp(-\frac{1}{2\sigma^2}(y- \mu)^2) \\ &= \exp(-\frac{1}{2\sigma^2}(y^2 - 2y \mu +\mu^2)- \frac{1}{2}\log(2\pi \sigma^2)) \\ &= \exp(\frac{y \mu - \mu^2/2}{\sigma^2} - \frac{y^2}{2\sigma^2} - \frac{1}{2}\log(2\pi \sigma^2)) \\ &= \exp(\frac{\theta y - b(\theta)}{a(\phi)} + c(y , \phi)) \end{aligned} $$

where

$\theta = \mu$
$b(\theta) = \frac{\mu^2}{2}$
$a(\phi) = \sigma^2 = \phi$
$c(y , \phi) = - \frac{1}{2}(\frac{y^2}{\phi}+\log(2\pi \sigma^2))$

Properties of GLM exponential families

$E(Y) = b' (\theta)$ where $b'(\theta) = \frac{\partial b(\theta)}{\partial \theta}$ (here ' is “prime”, not transpose)
$var(Y) = a(\phi)b''(\theta)= a(\phi)V(\mu)$.
- $V(\mu)$ is the variance function; however, it is only the variance in the case that $a(\phi) =1$
If $a(), b(), c()$ are identifiable, we will derive expected value and variance of Y.

Example

Normal distribution

$$ b’(\theta) = \frac{\partial b(\mu^2/2)}{\partial \mu} = \mu$$ $$V(\mu) = \frac{\partial^2 (\mu^2/2)}{\partial \mu^2} = 1$$ $$\to var(Y) = a(\phi) = \sigma^2 $$

Poisson distribution

$$ \begin{aligned} f(y, \theta, \phi) &= \frac{\mu^y \exp(-\mu)}{y!} &= \exp(y\log(\mu) - \mu - \log(y!)) &= \exp(y\theta - \exp(\theta) - \log(y!)) \end{aligned} $$

where

$\theta = \log(\mu)$
$a(\phi) = 1$
$b(\theta) = \exp(\theta)$
$c(y, \phi) = \log(y!)$

Hence,

$$ E(Y) = \frac{\partial b(\theta)}{\partial \theta} = \exp(\theta) = \mu$$ $$var(Y) = \frac{\partial^2 b(\theta)}{\partial \theta^2} = \mu $$

Since $\mu = E(Y) = b'(\theta)$

In GLM, we take some monotone function (typically nonlinear) of $\mu$ to be linear in the set of covariates

$$ g(\mu) = g(b’(\theta)) = \mathbf{x’\beta} $$

Equivalently,

$$ \mu = g^{-1}(\mathbf{x’\beta}) $$

where $g(.)$ is the link function since it links mean response ($\mu = E(Y)$) and a linear expression of the covariates

Some people use $\eta = \mathbf{x'\beta}$ where $\eta$ = the “linear predictor”

GLM is composed of 2 components

The random component:

is the distribution chosen to model the response variables $Y_1,...,Y_n$
is specified by the choice fo $a(), b(), c()$ in the exponential form
Notation:

Assume that there are n independent response variables $Y_1,...,Y_n$ with densities $$ f(y_i ; \theta_i, \phi) = \exp(\frac{\theta_i y_i - b(\theta_i)}{a(\phi)}+ c(y_i, \phi)) $$ notice each observation might have different densities - Assume that $\phi$ is constant for all $i = 1,...,n$, but $\theta_i$ will vary. $\mu_i = E(Y_i)$ for all i.

The systematic component

is the portion of the model that gives the relation between $\mu$ and the covariates $\mathbf{x}$
consists of 2 parts:
- the link function, $g(.)$
- the linear predictor, $\eta = \mathbf{x'\beta}$
Notation:

assume $g(\mu_i) = \mathbf{x'\beta} = \eta_i$ where $\mathbf{\beta} = (\beta_1,..., \beta_p)'$ - The parameters to be estimated are $\beta_1,...\beta_p , \phi$

The Canonical Link

To choose $g(.)$, we can use canonical link function (Remember: Canonical link is just a special case of the link function)

If the link function $g(.)$ is such $g(\mu_i) = \eta_i = \theta_i$, the natural parameter, then $g(.)$ is the canonical link.

Logistic Regression

$$ p_i = f(\mathbf{x}_i ; \beta) = \frac{exp(\mathbf{x_i’\beta})}{1 + exp(\mathbf{x_i’\beta})} $$

Equivalently,

$$ logit(p_i) = log(\frac{p_i}{1-p_i}) = \mathbf{x_i’\beta} $$

where $\frac{p_i}{1+p_i}$is the odds.

In this form, the model is specified such that a function of the mean response is linear. Hence, Generalized Linear Models

Logistic Regression: Interpretation of betas

For single regressor, the model is

$$ logit{\hat{p}_{x_i}} \equiv logit (\hat{p}_i) = \log(\frac{\hat{p}_i}{1 - \hat{p}_i}) = \hat{\beta}_0 + \hat{\beta}_1 x_i $$

When $x= x_i + 1$

$$ logit{\hat{{p}_{x_i +1}}} = \hat{\beta_0} + \hat{\beta}(x_i + 1)$$`

which is equal to $$logit\{\hat{p}_{x_i}\} + \hat{\beta}_1$$

Then,

$$ logit{\hat{p_{x_i +1}}} - logit{\hat{p_{x_i}}}$$

is equal to

$$log\{odds[\hat{p}_{x_i +1}]\} - log\{odds[\hat{p}_{x_i}]\}$$

the estimated odds ratio

the estimated odds ratio, when there is a difference of c units in the regressor x, is $exp(c\hat{\beta}_1)$. When there are multiple covariates, $exp(\hat{\beta}_k)$ is the estimated odds ratio for the variable $x_k$, assuming that all of the other variables are held constant.

Inference on the Mean Response*

Let $x_h = (1, x_{h1}, ...,x_{h,p-1})'$. Then

$$ \hat{p}_h = \frac{exp(\mathbf{x’_h \hat{\beta}})}{1 + exp(\mathbf{x’_h \hat{\beta}})} $$

and $s^2(\hat{p}_h) = \mathbf{x'_h[I(\hat{\beta})]^{-1}x_h}$

For new observation, we can have a cutoff point to decide whether y = 0 or 1.

setup

library(tidyverse)

df<-readxl::read_xlsx("logistic.xlsx") |> 
  mutate_if(is.character,as.factor) |> 
  rename(SEX="SEX JAN",
         STATUS="Default Status")
df
#> # A tibble: 6,824 x 7
#>    `RTGS EXPOSURE` STATUS SEX    `Marital Status` Profession   `Gross Income`
#>              <dbl>  <dbl> <fct>  <fct>            <fct>                 <dbl>
#>  1          1057.       1 MALE   Single           Blue Collar             652
#>  2           207.       1 MALE   Married          Blue Collar             460
#>  3          2162.       1 FEMALE Single           White Collar            468
#>  4            41.9      1 FEMALE Married          White Collar            733
#>  5            10.8      1 FEMALE Married          White Collar            465
#>  6           181.       1 FEMALE Married          White Collar            460
#>  7         14583.       0 MALE   Single           White Collar           8000
#>  8         53471.       0 FEMALE Other            Blue Collar            2600
#>  9          2156.       0 FEMALE Single           White Collar            773
#> 10          1114.       0 MALE   Married          Blue Collar             575
#> # i 6,814 more rows
#> # i 1 more variable: `Age Group` <fct>

CBZ is considering to develop a model that predicts the default experience of their loan books.A statistician was assigned to develop of prob of default,default and non-default,The factors that will be considered to influence.

EXPOSURE,SEX,MARIT,PROFFESSION,GROSSINCOME,AGEGROUP As predictor variables explaining all the terms

options(scipen = 999)
library(equatiomatic)
mod1<-glm(STATUS~.,data=df,family=binomial)
summary(mod1)
#> 
#> Call:
#> glm(formula = STATUS ~ ., family = binomial, data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.4612  -0.1916  -0.1331  -0.0539   4.1543  
#> 
#> Coefficients:
#>                              Estimate    Std. Error z value Pr(>|z|)    
#> (Intercept)               -2.13712297    1.23427306  -1.731   0.0834 .  
#> `RTGS EXPOSURE`           -0.00036209    0.00009165  -3.951 0.000078 ***
#> SEXMALE                   -0.01617771    0.27574625  -0.059   0.9532    
#> `Marital Status`Engaged  -12.87415038  514.79170656  -0.025   0.9800    
#> `Marital Status`Married   -0.91052261    1.04309052  -0.873   0.3827    
#> `Marital Status`Other     -2.28288908    1.44346105  -1.582   0.1138    
#> `Marital Status`Single    -1.05896113    1.06289344  -0.996   0.3191    
#> ProfessionWhite Collar    -0.32313990    0.26883397  -1.202   0.2294    
#> `Gross Income`            -0.00072287    0.00045865  -1.576   0.1150    
#> `Age Group`30-39           0.33735455    0.61753220   0.546   0.5849    
#> `Age Group`40-49           0.82290874    0.62040499   1.326   0.1847    
#> `Age Group`50-59           0.57181832    0.66677092   0.858   0.3911    
#> `Age Group`60-69           1.02085330    0.78462020   1.301   0.1932    
#> `Age Group`70+           -11.67550397 1624.62863041  -0.007   0.9943    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.30  on 6823  degrees of freedom
#> Residual deviance: 799.53  on 6810  degrees of freedom
#> AIC: 827.53
#> 
#> Number of Fisher Scoring iterations: 15

extract_eq(mod1,greek_colors = "blue",wrap=TRUE,terms_per_line = 2,use_coefs = TRUE)

null model

model with no dependent variables

model1<-glm(STATUS~1,data=df,family=binomial)
summary(model1)
#> 
#> Call:
#> glm(formula = STATUS ~ 1, family = binomial, data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.1545  -0.1545  -0.1545  -0.1545   2.9778  
#> 
#> Coefficients:
#>             Estimate Std. Error z value            Pr(>|z|)    
#> (Intercept)  -4.4218     0.1118  -39.56 <0.0000000000000002 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.3  on 6823  degrees of freedom
#> Residual deviance: 879.3  on 6823  degrees of freedom
#> AIC: 881.3
#> 
#> Number of Fisher Scoring iterations: 7

model with sex

model2<-glm(STATUS ~ SEX,data=df,family=binomial)
summary(model2)
#> 
#> Call:
#> glm(formula = STATUS ~ SEX, family = binomial, data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.1553  -0.1553  -0.1553  -0.1522   2.9882  
#> 
#> Coefficients:
#>             Estimate Std. Error z value            Pr(>|z|)    
#> (Intercept) -4.45312    0.23075 -19.299 <0.0000000000000002 ***
#> SEXMALE      0.04111    0.26376   0.156               0.876    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.30  on 6823  degrees of freedom
#> Residual deviance: 879.28  on 6822  degrees of freedom
#> AIC: 883.28
#> 
#> Number of Fisher Scoring iterations: 7

compare model with sex and null model

lrt<-anova(model1,model2,test="LRT")
lrt
#> Analysis of Deviance Table
#> 
#> Model 1: STATUS ~ 1
#> Model 2: STATUS ~ SEX
#>   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
#> 1      6823     879.30                     
#> 2      6822     879.28  1 0.024461   0.8757

model with marital status

model3<-glm(STATUS ~ `Marital Status`,data=df,family=binomial)
summary(model3)
#> 
#> Call:
#> glm(formula = STATUS ~ `Marital Status`, family = binomial, data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.2801  -0.1671  -0.1671  -0.1524   3.6591  
#> 
#> Coefficients:
#>                         Estimate Std. Error z value Pr(>|z|)   
#> (Intercept)               -3.219      1.020  -3.156   0.0016 **
#> `Marital Status`Engaged  -13.347    550.494  -0.024   0.9807   
#> `Marital Status`Married   -1.045      1.028  -1.017   0.3093   
#> `Marital Status`Other     -3.474      1.429  -2.432   0.0150 * 
#> `Marital Status`Single    -1.231      1.046  -1.177   0.2392   
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.3  on 6823  degrees of freedom
#> Residual deviance: 863.7  on 6819  degrees of freedom
#> AIC: 873.7
#> 
#> Number of Fisher Scoring iterations: 15

compare null model with marital status model

LRT1<-anova(model3,model1,test="LRT")
LRT1
#> Analysis of Deviance Table
#> 
#> Model 1: STATUS ~ `Marital Status`
#> Model 2: STATUS ~ 1
#>   Resid. Df Resid. Dev Df Deviance Pr(>Chi)   
#> 1      6819      863.7                        
#> 2      6823      879.3 -4  -15.607 0.003594 **
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

add exposure to marital status model

model5<-glm(STATUS ~ `RTGS EXPOSURE`+ `Marital Status`,data=df,family=binomial)

summary(model5)
#> 
#> Call:
#> glm(formula = STATUS ~ `RTGS EXPOSURE` + `Marital Status`, family = binomial, 
#>     data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.4322  -0.1971  -0.1385  -0.0604   4.3944  
#> 
#> Coefficients:
#>                             Estimate   Std. Error z value     Pr(>|z|)    
#> (Intercept)              -1.83322889   1.04911419  -1.747       0.0806 .  
#> `RTGS EXPOSURE`          -0.00044324   0.00008176  -5.421 0.0000000591 ***
#> `Marital Status`Engaged -13.08076745 523.70290688  -0.025       0.9801    
#> `Marital Status`Married  -0.99006551   1.03683577  -0.955       0.3396    
#> `Marital Status`Other    -2.34733702   1.43846067  -1.632       0.1027    
#> `Marital Status`Single   -1.24715662   1.05437281  -1.183       0.2369    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.30  on 6823  degrees of freedom
#> Residual deviance: 808.63  on 6818  degrees of freedom
#> AIC: 820.63
#> 
#> Number of Fisher Scoring iterations: 15

compare model with marital status to the one with both

LRT1<-anova(model3,model5,test="LRT")
LRT1
#> Analysis of Deviance Table
#> 
#> Model 1: STATUS ~ `Marital Status`
#> Model 2: STATUS ~ `RTGS EXPOSURE` + `Marital Status`
#>   Resid. Df Resid. Dev Df Deviance           Pr(>Chi)    
#> 1      6819     863.70                                   
#> 2      6818     808.63  1   55.066 0.0000000000001165 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

compare marital status model with model containing both

model6<-glm(STATUS ~ `RTGS EXPOSURE`+`Marital Status`+`Age Group`,data=df,family=binomial)
summary(model6)
#> 
#> Call:
#> glm(formula = STATUS ~ `RTGS EXPOSURE` + `Marital Status` + `Age Group`, 
#>     family = binomial, data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.4571  -0.1922  -0.1358  -0.0597   4.4447  
#> 
#> Coefficients:
#>                              Estimate    Std. Error z value     Pr(>|z|)    
#> (Intercept)               -2.42151338    1.20818555  -2.004        0.045 *  
#> `RTGS EXPOSURE`           -0.00044330    0.00008117  -5.461 0.0000000473 ***
#> `Marital Status`Engaged  -12.98152086  522.77176953  -0.025        0.980    
#> `Marital Status`Married   -0.88891522    1.03938005  -0.855        0.392    
#> `Marital Status`Other     -2.24263515    1.44042494  -1.557        0.119    
#> `Marital Status`Single    -1.04409472    1.06031177  -0.985        0.325    
#> `Age Group`30-39           0.26597436    0.61648455   0.431        0.666    
#> `Age Group`40-49           0.70606674    0.61702931   1.144        0.252    
#> `Age Group`50-59           0.40749938    0.66032219   0.617        0.537    
#> `Age Group`60-69           0.85164136    0.78032359   1.091        0.275    
#> `Age Group`70+           -11.67148243 1634.65825214  -0.007        0.994    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.30  on 6823  degrees of freedom
#> Residual deviance: 804.49  on 6813  degrees of freedom
#> AIC: 826.49
#> 
#> Number of Fisher Scoring iterations: 15

compare model with marital status to the one with both

LRT1<-anova(model5,model6,test="LRT")
LRT1
#> Analysis of Deviance Table
#> 
#> Model 1: STATUS ~ `RTGS EXPOSURE` + `Marital Status`
#> Model 2: STATUS ~ `RTGS EXPOSURE` + `Marital Status` + `Age Group`
#>   Resid. Df Resid. Dev Df Deviance Pr(>Chi)
#> 1      6818     808.63                     
#> 2      6813     804.49  5   4.1354   0.5301

age group is insignificant so we remove it

lets add Gross income

model7<-glm(STATUS ~ `RTGS EXPOSURE`+`Marital Status` + `Gross Income` ,data=df,family=binomial)
summary(model7)
#> 
#> Call:
#> glm(formula = STATUS ~ `RTGS EXPOSURE` + `Marital Status` + `Gross Income`, 
#>     family = binomial, data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.4207  -0.1971  -0.1374  -0.0558   4.1282  
#> 
#> Coefficients:
#>                             Estimate   Std. Error z value  Pr(>|z|)    
#> (Intercept)              -1.53001136   1.06871159  -1.432    0.1522    
#> `RTGS EXPOSURE`          -0.00036624   0.00009327  -3.927 0.0000861 ***
#> `Marital Status`Engaged -13.04375382 517.83889026  -0.025    0.9799    
#> `Marital Status`Married  -1.03211956   1.03805014  -0.994    0.3201    
#> `Marital Status`Other    -2.38777747   1.44020087  -1.658    0.0973 .  
#> `Marital Status`Single   -1.28721062   1.05555781  -1.219    0.2227    
#> `Gross Income`           -0.00070500   0.00046467  -1.517    0.1292    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.30  on 6823  degrees of freedom
#> Residual deviance: 805.54  on 6817  degrees of freedom
#> AIC: 819.54
#> 
#> Number of Fisher Scoring iterations: 15

compare model with gross income

LRT1<-anova(model5,model7,test="LRT")
LRT1
#> Analysis of Deviance Table
#> 
#> Model 1: STATUS ~ `RTGS EXPOSURE` + `Marital Status`
#> Model 2: STATUS ~ `RTGS EXPOSURE` + `Marital Status` + `Gross Income`
#>   Resid. Df Resid. Dev Df Deviance Pr(>Chi)  
#> 1      6818     808.63                       
#> 2      6817     805.54  1   3.0858  0.07898 .
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

final model should include gross income apparently
for sake of progress model with gross income is appropriate but we will use the one we used in the lecture

final model according to lecture

model7<-glm(STATUS ~ `RTGS EXPOSURE`+`Marital Status`,data=df,family=binomial)
summary(model7)
#> 
#> Call:
#> glm(formula = STATUS ~ `RTGS EXPOSURE` + `Marital Status`, family = binomial, 
#>     data = df)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -0.4322  -0.1971  -0.1385  -0.0604   4.3944  
#> 
#> Coefficients:
#>                             Estimate   Std. Error z value     Pr(>|z|)    
#> (Intercept)              -1.83322889   1.04911419  -1.747       0.0806 .  
#> `RTGS EXPOSURE`          -0.00044324   0.00008176  -5.421 0.0000000591 ***
#> `Marital Status`Engaged -13.08076745 523.70290688  -0.025       0.9801    
#> `Marital Status`Married  -0.99006551   1.03683577  -0.955       0.3396    
#> `Marital Status`Other    -2.34733702   1.43846067  -1.632       0.1027    
#> `Marital Status`Single   -1.24715662   1.05437281  -1.183       0.2369    
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 879.30  on 6823  degrees of freedom
#> Residual deviance: 808.63  on 6818  degrees of freedom
#> AIC: 820.63
#> 
#> Number of Fisher Scoring iterations: 15

calculation of probabilities

$$P_i=\frac{1}{1+e^-{(\beta_0+\sum^n_{i=1}\beta_iX_i)}}$$

e.g for client who is married with no information at the bank

1/(1+exp(-1.83-13.08076))
#> [1] 0.9999997

convert log odds to odds

coef(model7) |> exp()
#>             (Intercept)         `RTGS EXPOSURE` `Marital Status`Engaged 
#>            1.598964e-01            9.995569e-01            2.084946e-06 
#> `Marital Status`Married   `Marital Status`Other  `Marital Status`Single 
#>            3.715523e-01            9.562347e-02            2.873206e-01

reporting the model output

report::report(model7)
#> We fitted a logistic model (estimated using ML) to predict STATUS with RTGS
#> EXPOSURE and Marital Status (formula: STATUS ~ `RTGS EXPOSURE` + `Marital
#> Status`). The model's explanatory power is very weak (Tjur's R2 = 0.02). The
#> model's intercept, corresponding to RTGS EXPOSURE = 0 and Marital Status =
#> Divorced, is at -1.83 (95% CI [-4.74, -0.19], p = 0.081). Within this model:
#> 
#>   - The effect of RTGS EXPOSURE is statistically significant and negative (beta =
#> -4.43e-04, 95% CI [-6.14e-04, -2.93e-04], p < .001; Std. beta = -7.49, 95% CI
#> [-10.37, -4.96])
#>   - The effect of Marital Status [Engaged] is statistically non-significant and
#> negative (beta = -13.08, 95% CI [-202.23, 6.27], p = 0.980; Std. beta = -13.08,
#> 95% CI [-202.23, 6.27])
#>   - The effect of Marital Status [Married] is statistically non-significant and
#> negative (beta = -0.99, 95% CI [-2.60, 1.91], p = 0.340; Std. beta = -0.99, 95%
#> CI [-2.60, 1.91])
#>   - The effect of Marital Status [Other] is statistically non-significant and
#> negative (beta = -2.35, 95% CI [-5.61, 0.91], p = 0.103; Std. beta = -2.35, 95%
#> CI [-5.61, 0.91])
#>   - The effect of Marital Status [Single] is statistically non-significant and
#> negative (beta = -1.25, 95% CI [-2.91, 1.67], p = 0.237; Std. beta = -1.25, 95%
#> CI [-2.91, 1.67])
#> 
#> Standardized parameters were obtained by fitting the model on a standardized
#> version of the dataset. 95% Confidence Intervals (CIs) and p-values were
#> computed using a Wald z-distribution approximation.

Bongani Ncube

Data Scientist/Statistics/Public Health

My research interests include Casual Inference , Public Health , Survival Analysis, bayesian Statistics, Machine learning and Longitudinal Data Analysis.