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 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.
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.
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) head(param_grid)
# 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
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))
}
# Initial values (adjust based on your data if necessary)
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
)
}
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…
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.estimate_chisq_params demonstrates the most accurate and unbiased estimates across different df values.ncp Estimation:
mean(x) - mean(x) - 1 and mean(x) - var(x) / 2) show substantial bias and variability.optim) performs better but still exhibits some bias.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!