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As the degrees of freedom increase, the chi-square curve approaches a normal distribution. But what about increasing the sample size? If I am using degree of freedom =1, but with different sample sizes (eg n=10, 30 and 50), will the chi-square curve become more normal with increasing sample size?

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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Dec 30, 2022 at 8:24
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    $\begingroup$ The $\chi^2(d)$ distribution is a different distribution from the normal. A completely different family. It is also an idealized mathematical distribution of what is supposed to happen. In other words, as $n\to \infty$ (your sample size), then the $\chi^2(d)$ statistic that you calculate, will be shaped at that idealized distribution. To put it simply, as your increase the sample size, then the distribution of statistic values you get will converge to that chi-squared one. $\endgroup$ Dec 30, 2022 at 8:54
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    $\begingroup$ How do you increase the sample size without increasing the degrees of freedom? Are you just drawing more and more samples from a $\chi^2(1)$ distribution? If so, why would you think a histogram or curve would approach anything other than a $\chi^2(1)$ distribution? $\endgroup$
    – jbowman
    Dec 30, 2022 at 16:18
  • $\begingroup$ Are you asking about the situation in a goodness of fit test or perhaps an independence test where the number of categories is held constant but the total sample size increases? $\endgroup$
    – Glen_b
    Dec 30, 2022 at 23:28

2 Answers 2

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Assuming that you are asking about the situation in a goodness of fit test or perhaps an independence test where the number of categories is held constant but the total sample size increases:

The chi-squared curve doesn't become more normal; the d.f. is unchanged.

What's changing is how well the cell counts are approximated by a normal distribution. More accurately, what's changing is how well suitably scaled cell counts are approximated by a (typically degenerate) multivariate normal distribution, and hence how well the discrete test statistic $\sum_i\frac{(O_i-E_i)^2}{E_i}$ is approximated by a (continuous) chi squared distribution.

It's this approximation that leads to rules like "expected counts >5" in an attempt to get reasonable accuracy of significance levels across a range of possible situations (different hypothesized $p_i$ in a goodness of fit test). The rule is often 'too strict', but occasionally not really strict enough (at least for my preferences of accuracy of p-values).

Example of improving approximation with sample size:

Plot of chi-squared(2) approximation compared to actual distribution of a chi-squared goodness of fit test statistic at n=15 and n=50
(You can see a larger image by clicking on the plot, or right-clicking to open the plot in a new tab)

The plot on the left shows the distribution function of a Pearson chi-squared goodness of fit test for three groups with null cell probabilities $0.2, 0.3, 0.5$ for a sample of size $n=15$ (black step function) and the continuous approximation by a chi-squared(2) distribution in red. Here the smallest expected count is 3.

This case was not chosen to show a situation where the approximation is particularly poor; its a 'nice' case. The broad picture looks reasonably close but consider that the exact 5% critical value is 5.756 while the chi-squared approximation gives 5.991, so there is some error, though it's not especially impactful if you do your test at the 5% level in this case, in that a 5% test is mildly conservative.

The approximation error has a somewhat larger impact at the 10% level for this specific case.

The 'exact' cdf is simulated (calculated by sampling from its distribution under $H_0$), but the simulation size is such that the worst-case margin of error in the estimate of the cdf is only a little over a pixel (which occurs near the median; near the 5% critical value it's nearer to half a pixel), so the picture is effectively exact; the differences you see are real, not due to sampling error -- if you did a new simulation of the same size the plot would be indistinguishable from this one.

The plot on the right is similar but for n=50 (smallest expected count is 10); as we see, the chi-squared approximation is considerably better.

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One of the best ways to understand statistical concepts is just to simulate the random experiments yourself. Here is some R code which illustrates what happens as you increase the sample size.

simulate = function(n){
  data = rpois(n, lambda = 1)
  Obs.0 = sum(data == 0)
  Obs.1 = sum(data == 1)
  Obs.2 = sum(data >= 2)
  Exp.0 = dpois(0, lambda = 1)*n
  Exp.1 = dpois(1, lambda = 1)*n
  Exp.2 = n - (Exp.0 + Exp.1)
  chi = 0
  chi = chi + (Obs.0-Exp.0)^2/Exp.0
  chi = chi + (Obs.1-Exp.1)^2/Exp.1
  chi = chi + (Obs.2-Exp.2)^2/Exp.2
  chi
}

result = replicate(1e2, simulate(1e2))
plot(density(result))
curve(dchisq(x,2), add=TRUE, col="red")

Let us explain what this code does. When you run simulate(n), the computer generates $n$ (independent) random samples from Poisson(1). We created three categories, whose where the sample = 0, those were sample = 1, and those were sample >= 2. We calculate the expected count vs the actual count. In the end the chi-squared statistic is calculated for that data set.

But this is just 1 number. So now we replicate this process $10^2$ times. Generating $10^2$ different values of the chi-squared statistic. Then we tell the computer to draw the estimate distribution function for all those chi-squared values. Finally, the idealized chi-squared distribution is drawn in red.

Notice, if you increase the replication number from $10^2$ slowly up to $10^5$, the fit gets closer to the idealized chi-squared.

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