# Statistics 5601 (Geyer, Fall 2003) Examples: One-Way Layout

## General Instructions

To do each example, just click the "Submit" button. You do not have to type in any R instructions or specify a dataset. That's already done for you.

## Kruskal-Wallis

### Example 6.1 in Hollander and Wolfe.

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The second analysis done by the `aov` function is the usual parametric procedure: one-way ANOVA. It produces P = 0.5866 for comparison with the Kruskal-Wallis P-value.

## Jonckheere-Terpstra

Unfortunately, R doesn't have this procedure. So we'll have to do it by hand (in R).

### Example 6.2 in Hollander and Wolfe.

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### Summary

• Upper-tailed Jonckheere-Terpstra test
• Test statistic: J = 79
• Sample sizes: n1 = 6, n2 = 6, and n3 = 6
• Monte Carlo approximation to P-value: P = 0.0231
• Monte Carlo standard error of P-value: 0.0015

### Comment

Rather than use large sample approximation on what are really small sample sizes, we do a Monte Carlo calculation of the P-value (that is, we compute by simulation of null distribution of the test statistic).

The Monte Carlo calculation is the loop

```for (i in 1:nsim) {
datsim <- sample(dat, length(dat))
jsim[i] <- jkstat(datsim, grp)
}
```

This does `nsim` simulations of the null distribution of the test statistic. The first line of the body of the loop generates a new simulated data set `datsim` which is a permutation of the actual data (same numbers, just assigned to different groups). The second line of the body of the loop calculates the value of the test statistic for the simulated data and stores it for future use.

After the loop has finished `jsim` is a vector of length `nsim` that consists of independent, identically distributed random variables having the distribution of the test statistic J under the null hypothesis. And

```phat <- mean(jsim >= jstat)
```

approximates the P-value, which is Pr(Jj).

The slightly more tricky code

```(nsim * phat + 1) / (nsim + 1)
```

includes the observed value in the numerator and denominator. As explained in class, this assures that if α is a multiple of `1 / nsim`, then Pr(P ≤ α) is indeed α, despite the Monte Carlo.

Despite having an exact Monte Carlo test (exact meaning level α really means level α), there is some interest in the randomness in the reported P-value. Hence the next to last line calculates its standard error.

The last line reports the time the calculation takes: the first number is the elapsed time in seconds.

## Isotonic Regression

This is the normal-theory competitor to Jonckheere-Terpstra. Unfortunately, R doesn't have this procedure. So we'll have to do it by hand (in R).

### Example 6.2 in Hollander and Wolfe.

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### Summary

• Upper-tailed isotonic regression test
• Test statistic: T = 52.78
• Sample sizes: n1 = 6, n2 = 6, and n3 = 6
• Monte Carlo approximation to P-value: P = 0.036
• Monte Carlo standard error of P-value: 0.0019

### Comment

The result quoted in the summary uses 10 times the sample size entered in the example form above. It was run off-line (R rather than Rweb) with the following output.

The R function `pava` implements the pool adjacent violators algorithm which does isotonic regression. In the famous words of a UNIX source code comment you are not expected to understand this.

The main lesson here is that the normal theory test (isotonic regression) does more or less the same as the nonparametric Jonckheere-Terpstra test. This is no surprise since the data are fairly normal looking.

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