# Estimating Chi-Square Distribution Parameters Using R

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rtip
operations
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unglue
Author

Steven P. Sanderson II, MPH

Published

April 15, 2024

# Introduction

In the world of statistics and data analysis, understanding and accurately estimating the parameters of probability distributions is crucial. One such distribution is the chi-square distribution, often encountered in various statistical analyses. In this blog post, we’ll dive into how we can estimate the degrees of freedom (“df”) and the non-centrality parameter (“ncp”) of a chi-square distribution using R programming language.

# The Chi-Square Distribution

The chi-square distribution is a continuous probability distribution that arises in the context of hypothesis testing and confidence interval estimation. It is commonly used in goodness-of-fit tests, tests of independence, and tests of homogeneity.

The distribution has two main parameters: - Degrees of Freedom (df): This parameter determines the shape of the chi-square distribution. It represents the number of independent variables in a statistical test. - Non-Centrality Parameter (ncp): This parameter determines the deviation of the distribution from a null hypothesis. It’s particularly relevant in non-central chi-square distributions.

# The Goal: Estimating Parameters

Our goal is to create a function within the TidyDensity package that can estimate the df and ncp parameters of a chi-square distribution based on a vector of observed data. Let’s walk through the steps involved in achieving this.

# Working Example

## Setting the Stage: Libraries and Data

First, we load the necessary libraries: `tidyverse` for data manipulation and `bbmle` for maximum likelihood estimation. We then generate a grid of parameters (degrees of freedom and non-centrality parameter) and sample sizes to create a diverse set of chi-square distributed data.

``````# Load libraries
library(tidyverse)
library(bbmle)

# Data ----
# Make parameters and grid
df <- 1:10
ncp <- 1:10
n <- runif(10, 250, 500) |> trunc()
param_grid <- expand_grid(n = n, df = df, ncp = ncp)

``````# A tibble: 6 × 3
n    df   ncp
<dbl> <int> <int>
1   284     1     1
2   284     1     2
3   284     1     3
4   284     1     4
5   284     1     5
6   284     1     6``````

## Function Exploration: Unveiling the Estimation Process

The core of our exploration lies in several functions designed to estimate the chi-square parameters:

`dof`/`k` Functions: These functions focus on estimating the degrees of freedom (df) using different approaches:

• `mean_x`: Calculates the mean of the data.
• `mean_minus_1`: Subtracts 1 from the mean.
• `var_div_2`: Divides the variance of the data by 2.
• `length_minus_1`: Subtracts 1 from the length of the data.

`ncp` Functions: These functions aim to estimate the non-centrality parameter (ncp) using various methods:

• `mean_minus_mean_minus_1`: A seemingly trivial calculation that serves as a baseline.
• `ie_mean_minus_var_div_2`: Subtracts half the variance from the mean, ensuring the result is non-negative.
• `ie_optim`: Utilizes optimization techniques to find the ncp that maximizes the likelihood of observing the data.
• `estimate_chisq_params`: This is the main function that employs maximum likelihood estimation (MLE) via the bbmle package to estimate both df and ncp simultaneously. It defines a negative log-likelihood function based on the chi-square distribution and uses mle2 to find the parameter values that minimize this function.
``````# Functions ----
# functions to estimate the parameters of a chisq distribution
# dof
mean_x <- function(x) mean(x)
mean_minus_1 <- function(x) mean(x) - 1
var_div_2 <- function(x) var(x) / 2
length_minus_1 <- function(x) length(x) - 1
# ncp
mean_minus_mean_minus_1 <- function(x) mean(x) - (mean(x) - 1)
ie_mean_minus_var_div_2 <- function(x) ifelse((mean(x) - (var(x) / 2)) < 0, 0, mean(x) - var(x)/2)
ie_optim <- function(x) optim(par = 0,
fn = function(ncp) {
-sum(dchisq(x, df = var(x)/2, ncp = ncp, log = TRUE))
},
method = "Brent",
lower = 0,
upper = 10 * var(x)/2)\$par
# both
estimate_chisq_params <- function(data) {
# Negative log-likelihood function
negLogLik <- function(df, ncp) {
-sum(dchisq(data, df = df, ncp = ncp, log = TRUE))
}

start_vals <- list(df = trunc(var(data)/2), ncp = trunc(mean(data)))

# MLE using bbmle
mle_fit <- bbmle::mle2(negLogLik, start = start_vals)
# Return estimated parameters as a named vector
df <- dplyr::tibble(
est_df = coef(mle_fit)[1],
est_ncp = coef(mle_fit)[2]
)
return(df)
}

safe_estimates <- {
purrr::possibly(
estimate_chisq_params,
otherwise = NA_real_,
quiet = TRUE
)
}``````

## Simulating and Evaluating: Putting the Functions to the Test

To assess the performance of our functions, we simulate chi-square data using the parameter grid and apply each function to estimate the parameters. We then compare these estimates to the true values and visualize the results using boxplots.

``````# Simulate data ----
set.seed(123)
dff <- param_grid |>
mutate(x = pmap(pick(everything()), match.fun("rchisq"))) |>
mutate(
safe_est_parms = map(x, safe_estimates),
dfa = map_dbl(x, mean_minus_1),
dfb = map_dbl(x, var_div_2),
dfc = map_dbl(x, length_minus_1),
ncpa = map_dbl(x, mean_minus_mean_minus_1),
ncpb = map_dbl(x, ie_mean_minus_var_div_2),
ncpc = map_dbl(x, ie_optim)
) |>
select(-x) |>
filter(map_lgl(safe_est_parms, ~ any(is.na(.x))) == FALSE) |>
unnest(cols = safe_est_parms) |>
mutate(
dfa_resid = dfa - df,
dfb_resid = dfb - df,
dfc_resid = dfc - df,
dfd_resid = est_df - df,
ncpa_resid = ncpa - ncp,
ncpb_resid = ncpb - ncp,
ncpc_resid = ncpc - ncp,
ncpd_resid = est_ncp - ncp
)

glimpse(dff)``````
``````Rows: 987
Columns: 19
\$ n          <dbl> 284, 284, 284, 284, 284, 284, 284, 284, 284, 284, 284, 284,…
\$ df         <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,…
\$ ncp        <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1…
\$ est_df     <dbl> 1.1770904, 0.9905994, 0.9792179, 0.7781877, 1.5161669, 0.82…
\$ est_ncp    <dbl> 0.7231638, 1.9462325, 3.0371756, 4.2347494, 3.7611119, 6.26…
\$ dfa        <dbl> 0.9050589, 1.9826153, 3.0579375, 4.0515312, 4.2022289, 6.15…
\$ dfb        <dbl> 2.626501, 5.428382, 7.297746, 9.265272, 8.465838, 14.597976…
\$ dfc        <dbl> 283, 283, 283, 283, 283, 283, 283, 283, 283, 283, 283, 283,…
\$ ncpa       <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
\$ ncpb       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
\$ ncpc       <dbl> 5.382789e-09, 8.170550e-09, 6.017177e-09, 8.618892e-09, 7.7…
\$ dfa_resid  <dbl> -0.09494109, 0.98261533, 2.05793748, 3.05153121, 3.20222890…
\$ dfb_resid  <dbl> 1.626501, 4.428382, 6.297746, 8.265272, 7.465838, 13.597976…
\$ dfc_resid  <dbl> 282, 282, 282, 282, 282, 282, 282, 282, 282, 282, 281, 281,…
\$ dfd_resid  <dbl> 0.177090434, -0.009400632, -0.020782073, -0.221812344, 0.51…
\$ ncpa_resid <dbl> 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, 0, -1, -2, -3, -4, -…
\$ ncpb_resid <dbl> -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -1, -2, -3, -4, -5…
\$ ncpc_resid <dbl> -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -1, -2, -3, -4, -5…
\$ ncpd_resid <dbl> -0.27683618, -0.05376753, 0.03717560, 0.23474943, -1.238888…``````

## Visual Insights: Assessing Estimation Accuracy

The boxplots reveal interesting insights:

``````par(mfrow = c(1, 2))
boxplot(dff\$dfa ~ dff\$df, main = "mean(x) -1 ~ df")
boxplot(dff\$dfa_resid ~ dff\$df, main = "mean(x) -1 ~ df Residuals")``````

``````par(mfrow = c(1, 1))

par(mfrow = c(1, 2))
boxplot(dff\$dfb ~ dff\$df, main = "var(x) / 2 ~ df")
boxplot(dff\$dfb_resid ~ dff\$df, main = "var(x) / 2 ~ df Residuals")``````

``````par(mfrow = c(1, 1))

par(mfrow = c(1, 2))
boxplot(dff\$dfc ~ dff\$df, main = "length(x) - 1 ~ df")
boxplot(dff\$dfc_resid ~ dff\$df, main = "length(x) - 1 ~ df Residuals")``````

``````par(mfrow = c(1, 1))

par(mfrow = c(1, 2))
boxplot(dff\$est_df ~ dff\$df, main = "negloglik ~ df - Looks Good")
boxplot(dff\$dfd_resid ~ dff\$df, main = "negloglik ~ df Residuals")``````

``````par(mfrow = c(1, 1))

par(mfrow = c(1, 2))
boxplot(dff\$ncpa ~ dff\$ncp, main = "mean(x) - (mean(x) - 1) ~ ncp")
boxplot(dff\$ncpa_resid ~ dff\$ncp, main = "mean(x) - (mean(x) - 1) ~ ncp Residuals")``````

``````par(mfrow = c(1, 1))

par(mfrow = c(1, 2))
boxplot(dff\$ncpb ~ dff\$ncp, main = "mean(x) - var(x)/2 ~ nc")
boxplot(dff\$ncpb_resid ~ dff\$ncp, main = "mean(x) - var(x)/2 ~ ncp Residuals")``````

``````par(mfrow = c(1, 1))

par(mfrow = c(1, 2))
boxplot(dff\$ncpc ~ dff\$ncp, main = "optim ~ ncp")
boxplot(dff\$ncpc_resid ~ dff\$ncp, main = "optim ~ ncp Residuals")``````

``````par(mfrow = c(1, 1))

par(mfrow = c(1, 2))
boxplot(dff\$est_ncp ~ dff\$ncp, main = "negloglik ~ ncp - Looks Good")
boxplot(dff\$ncpd_resid ~ dff\$ncp, main = "negloglik ~ ncp Residuals")``````

``par(mfrow = c(1, 1))``

`df` Estimation:

• `mean_x - 1 and var(x) / 2` show potential as df estimators but exhibit bias depending on the true df value.
• `length(x) - 1` performs poorly, consistently underestimating df.
• The MLE approach from `estimate_chisq_params` demonstrates the most accurate and unbiased estimates across different df values.

`ncp` Estimation:

• The simple methods (`mean(x) - mean(x) - 1` and `mean(x) - var(x) / 2`) show substantial bias and variability.
• The optimization-based method (`optim`) performs better but still exhibits some bias.
• The MLE approach again emerges as the most reliable option, providing accurate and unbiased estimates across various ncp values.

# Conclusion: The Power of Maximum Likelihood

Our exploration highlights the effectiveness of MLE in estimating the parameters of a chi-square distribution. The estimate_chisq_params function, utilizing the bbmle package, provides a robust and accurate solution for this task. This function will be a valuable addition to the TidyDensity package, empowering users to delve deeper into the analysis of chi-square distributed data.

Stay tuned for further developments and exciting additions to the TidyDensity package!