AE 07: Newton Raphson

Important

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Render, commit, and push your responses to GitHub by the end of class to submit your AE.

library(tidyverse)
library(knitr)
library(tidymodels)

Introduction

COVID-19 infection prevention practices at food establishments

Researchers at Wollo University in Ethiopia conducted a study in July and August 2020 to understand factors associated with good COVID-19 infection prevention practices at food establishments. Their study is published in Andualem et al. (2022).

They were particularly interested in the understanding implementation of prevention practices at food establishments, given the workers’ increased risk due to daily contact with customers.

We will use the data from Andualem et al. (2022) to explore the association between age, sex, years of service, and whether someone works at a food establishment with access to personal protective equipment (PPE) as of August 2020. We will use access to PPE as a proxy for wearing PPE.

The study participants were selected using a simple random sampling at the selected establishments.

.

covid_df <- read_csv("data/covid-prevention-study.csv") |>
  rename(age = "Age of food handlers", 
         years = "Years of service", 
         ppe_access = "Availability of PPEs") |>
  mutate(sex = factor(if_else(Sex == 2, "Female", "Male")),
         ppe_access = as_factor(ppe_access))

Functions for Newton-Raphson

## Calculate the first derivative of logL (score function)

calc_first_deriv <- function(beta, X, y){
  first_deriv <- rep(0, length(beta))
  for(i in 1:length(y)){
    first_deriv <- first_deriv + as.numeric(y[i] - exp(X[i,] %*% beta)/(1 + exp(X[i,] %*% beta))) %*% X[i,]
  }
  return(colSums(first_deriv)) #return values as a vector 
}

## Calculate the second derivative of logL (Hessian)

calc_second_deriv <- function(beta, X, y){
  second_deriv <- matrix(0, nrow = length(beta), ncol = length(beta))
  for(i in 1:length(y)){
    second_deriv <- second_deriv + as.numeric(1/(1 + exp(X[i,] %*% beta)))*
      as.numeric((exp(X[i,] %*% beta)/(1 + exp(X[i,] %*% beta)))) * 
      (X[i,]) %*% t(X[i,])
  }
  return(second_deriv)
}

Get starting values

# design matrix
X <- model.matrix(~ age + sex + years, data = covid_df)

# vector of response
y <- I(covid_df$ppe_access == 1)

# vector of coefficients
beta <- c(0, 0, 0, 0)

# keep track of iterations
iter <- 1

# keep track of difference in estimates
delta <- 1

# keep track of estimates at each iteration 
temp <- matrix(0, nrow = 500, ncol = 4) 

Estimate $\mathit{\beta }$

while(delta > 0.000001 & iter < 50){
  old <- beta
  beta <- old - solve(-1 * calc_second_deriv(beta = beta, X = X, y = y)) %*% 
                calc_first_deriv(beta = beta, X = X, y = y)
  temp[iter,] <- beta
  delta <- sqrt(sum((beta - old)^2))
  iter <- iter + 1
}

Show results

iter

[1] 7

beta

                   [,1]
(Intercept) -2.12709823
age          0.05586608
sexMale      0.34070061
years        0.26396205

Note that $({\mathrm{\nabla }}_{\mathit{\beta }}^2\log L{)}^{-1}$is the estimate for $Var(\hat{\mathit{\beta }})$ . The square root of the diagonal elements are the estimates for $SE(\hat{\mathit{\beta }})$.

#calculate the hessian matrix
second_deriv <- calc_second_deriv(beta = beta, X = X, y = y)
  
# take the inverse
inv_second_deriv <- solve(second_deriv)

# get estimates for SE
se_beta <- sqrt(diag(inv_second_deriv))

se_beta

[1] 0.45832580 0.01740581 0.22360561 0.06583117

Coefficient estimates from glm

ppe_model <- glm(ppe_access ~ age + sex + years, 
                 data = covid_df, family = binomial)
tidy(ppe_model, conf.int = TRUE) |>
  kable(digits = 3)

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	-2.127	0.458	-4.641	0.000	-3.058	-1.257
age	0.056	0.017	3.210	0.001	0.023	0.091
sexMale	0.341	0.224	1.524	0.128	-0.098	0.780
years	0.264	0.066	4.010	0.000	0.143	0.401

References

Andualem, Atsedemariam, Belachew Tegegne, Sewunet Ademe, Tarikuwa Natnael, Gete Berihun, Masresha Abebe, Yeshiwork Alemnew, et al. 2022. “COVID-19 Infection Prevention Practices Among a Sample of Food Handlers of Food and Drink Establishments in Ethiopia.” Edited by Ammal Mokhtar Metwally. PLOS ONE 17 (1): e0259851. https://doi.org/10.1371/journal.pone.0259851.