# Chi-square Goodness of Fit Test in R

# What is chi-square goodness of fit test?

**chi-square**

**goodness of fit**test is used to compare the observed distribution to an expected distribution, in a situation where we have two or more categories in a discrete data. In other words, it compares multiple observed proportions to expected probabilities.

# Example data and questions

For example, we collected wild tulips and found that 81 were red, 50 were yellow and 27 were white.

**Question 1**:

Are these colors equally common?

If these colors were equally distributed, the expected proportion would be 1/3 for each of the color.

**Question 2**:

Suppose that, in the region where you collected the data, the ratio of red, yellow and white tulip is 3:2:1 (3+2+1 = 6). This means that the expected proportion is:

- 3/6 (= 1/2) for red
- 2/6 ( = 1/3) for yellow
- 1/6 for white

We want to know, if there is any significant difference between the observed proportions and the expected proportions.

# Statistical hypotheses

*Null hypothesis*(\(H_0\)): There is no significant difference between the observed and the expected value.*Alternative hypothesis*(\(H_a\)): There is a significant difference between the observed and the expected value.

# R function: chisq.test()

The R function **chisq.test**() can be used as follow:

`chisq.test(x, p)`

**x**: a numeric vector**p**: a vector of probabilities of the same length of x.

## Answer to Q1: Are the colors equally common?

```
tulip <- c(81, 50, 27)
res <- chisq.test(tulip, p = c(1/3, 1/3, 1/3))
res
```

```
Chi-squared test for given probabilities
data: tulip
X-squared = 27.886, df = 2, p-value = 8.803e-07
```

The **p-value** of the test is 8.80310^{-7}, which is less than the significance level alpha = 0.05. We can conclude that the colors are significantly not commonly distributed with a **p-value** = 8.80310^{-7}.

Note that, the chi-square test should be used only when all calculated expected values are greater than 5.

```
# Access to the expected values
res$expected
```

`[1] 52.66667 52.66667 52.66667`

## Answer to Q2 comparing observed to expected proportions

```
tulip <- c(81, 50, 27)
res <- chisq.test(tulip, p = c(1/2, 1/3, 1/6))
res
```

```
Chi-squared test for given probabilities
data: tulip
X-squared = 0.20253, df = 2, p-value = 0.9037
```

The **p-value** of the test is 0.9037, which is greater than the significance level alpha = 0.05. We can conclude that the observed proportions are not significantly different from the expected proportions.

## Access to the values returned by chisq.test() function

The result of **chisq.test()** function is a list containing the following components:

**statistic**: the value the chi-squared test statistic.**parameter**: the degrees of freedom**p.value**: the**p-value**of the test**observed**: the observed count**expected**: the expected count

The format of the **R** code to use for getting these values is as follow:

```
# printing the p-value
res$p.value
```

`[1] 0.9036928`

```
# printing the mean
res$estimate
```

`NULL`

# See also

# Infos

This analysis has been performed using **R software** (ver. 3.2.4).

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