Chi squared test

Klinkenberg

University of Amsterdam

22 oct 2022

\(\chi^2\) test

Relation between categorical variables

\(\chi^2\) test

A ’‘’chi-squared test’’‘, also written as \(\chi^2\) test, is any statistical hypothesis test wherein the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Without other qualification, ’chi-squared test’ often is used as short for Pearson’s chi-squared test.

Chi-squared tests are often constructed from a Lack-of-fit sum of squares#Sums of squares|sum of squared errors, or through the Variance Distribution of the sample variance|sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent.

Source: wikipedia

\(\chi^2\) test statistic

\(\chi^2 = \sum \frac{(\text{observed}_{ij} - \text{model}_{ij})^2}{\text{model}_{ij}}\)

Contingency table

\(\text{observed}_{ij} = \begin{pmatrix} o_{11} & o_{12} & \cdots & o_{1j} \\ o_{21} & o_{22} & \cdots & o_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ o_{i1} & o_{i2} & \cdots & o_{ij} \end{pmatrix}\)

\(\text{model}_{ij} = \begin{pmatrix} m_{11} & m_{12} & \cdots & m_{1j} \\ m_{21} & m_{22} & \cdots & m_{2j} \\ \vdots & \vdots & \ddots & \vdots \\ m_{i1} & m_{i2} & \cdots & m_{ij} \end{pmatrix}\)

\(\chi^2\) distribution

The \(\chi^2\) distribution describes the test statistic under the assumption of \(H_0\), given the degrees of freedom.

\(df = (r - 1) (c - 1)\) where \(r\) is the number of rows and \(c\) the amount of columns.

chi = seq(0,8,.01)
df  = c(1,2,3,6,8,10)
col = rainbow(n = length(df))

plot( chi, dchisq(chi, df[1]), lwd = 2, col = col[1], type="l",
      main = "Chi squared distributions",
      ylab = "Density",
      ylim = c(0,1),
      xlab = "Chi squared")

lines(chi, dchisq(chi, df[2]), lwd = 2, col = col[2], type="l")
lines(chi, dchisq(chi, df[3]), lwd = 2, col = col[3], type="l")
lines(chi, dchisq(chi, df[4]), lwd = 2, col = col[4], type="l")
lines(chi, dchisq(chi, df[5]), lwd = 2, col = col[5], type="l")
lines(chi, dchisq(chi, df[6]), lwd = 2, col = col[6], type="l")

legend("topright", legend = paste("k =",df), col = col, lty = 1, bty = "n")

\(\chi^2\) distribution

Example

Simon

Experiment

http://goo.gl/faj76B

Data

Calculating \(\chi^2\)

observed <- table(results[,c("fluiten","sekse")])
observed

       sekse
fluiten Man Vrouw
    Ja   17    26
    Nee   1    12

\(\text{observed}_{ij} = \begin{pmatrix} 17 & 26 \\ 1 & 12 \\ \end{pmatrix}\)

Calculating the model

\(\text{model}_{ij} = E_{ij} = \frac{\text{row total}_i \times \text{column total}_j}{n }\)

n   = sum(observed)
ct1 = colSums(observed)[1]
ct2 = colSums(observed)[2]
rt1 = rowSums(observed)[1]
rt2 = rowSums(observed)[2]

addmargins(observed)

       sekse
fluiten Man Vrouw Sum
    Ja   17    26  43
    Nee   1    12  13
    Sum  18    38  56

Calculating the model

\(\text{model}_{ij} = E_{ij} = \frac{\text{row total}_i \times \text{column total}_j}{n }\)

model = matrix( c((ct1*rt1)/n,
                  (ct2*rt1)/n,
                  (ct1*rt2)/n,
                  (ct2*rt2)/n),2,2,byrow=T
          )
model

          [,1]      [,2]
[1,] 13.821429 29.178571
[2,]  4.178571  8.821429

\(\text{model}_{ij} = \begin{pmatrix} 13.8214286 & 29.1785714 \\ 4.1785714 & 8.8214286 \\ \end{pmatrix}\)

observed - model

observed - model

       sekse
fluiten       Man     Vrouw
    Ja   3.178571 -3.178571
    Nee -3.178571  3.178571

Calculating \(\chi^2\)

\(\chi^2 = \sum \frac{(\text{observed}_{ij} - \text{model}_{ij})^2}{\text{model}_{ij}}\)

# Calculate chi squared
chi.squared <- sum((observed - model)^2/model)
chi.squared

[1] 4.64045

Testing for significance

\(df = (r - 1) (c - 1)\)

df = (2 - 1) * ( 2 - 1)

library('visualize')
visualize.chisq(chi.squared,df,section='upper')

Testing for significance

Fisher’s exact test

Calculates axact \(\chi^2\) for small samples.

Cell size < 5

Yates’s correction

For 2 x 2 contingency tables.

\(\chi^2 = \sum \frac{ ( | \text{observed}_{ij} - \text{model}_{ij} | - .5)^2}{\text{model}_{ij}}\)

# Calculate Yates's corrected chi squared
chi.squared.yates <- sum((abs(observed - model) - .5)^2/model)
chi.squared.yates

Yates’s correction

[1] 3.295358

visualize.chisq(chi.squared.yates,df,section='upper')

Likelihood ratio

Alternatieve to Pearson’s \(\chi^2\).

\(L \chi^2 = 2 \sum \text{observed}_{ij} ln \left( \frac{\text{observed}_{ij}}{\text{model}_{ij}} \right)\)

# ln is log
lx2 = 2 * sum(observed * log(observed / model) ); lx2

Likelihood ratio

[1] 5.565598

visualize.chisq(lx2,df,section='upper')

Standardized residuals

\(\text{standardized residuals} = \frac{ \text{observed}_{ij} - \text{model}_{ij} }{ \sqrt{ \text{model}_{ij} } }\)

(observed - model) / sqrt(model)

       sekse
fluiten        Man      Vrouw
    Ja   0.8549791 -0.5884370
    Nee -1.5549558  1.0701940

Effect size

Odds ratio based on the observed values

odds <- round( observed, 2); odds

       sekse
fluiten Man Vrouw
    Ja   17    26
    Nee   1    12

\(\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\)

\(OR = \frac{a \times d}{b \times c} = \frac{17 \times 12}{26 \times 1} = 7.8461538\)

Odds

       sekse
fluiten Man Vrouw
    Ja   17    26
    Nee   1    12

The man and women ratio of people that can whisle and the ratio of those who can’t whistle

Can wistle \(\text{Odds}_{mf} = \frac{ 17 }{ 26 }\) = 0.6538462
Can’t wistle \(\text{Odds}_{mf} = \frac{ 1 }{ 12 }\) = 0.0833333

Odds ratio

Is the ratio of these odds.

\(OR = \frac{\text{wistle}}{\text{can't wistle}} = \frac{0.6538462}{0.0833333} = 7.8461538\)

End

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