Topic 12

Question 1

Why might we use a Chi square test instead of a T test

Answer 1

This is because the other tests such as sample T tests only measure quantitative data, whereas Chi square tests are able to measure qualitative data. Additionally, it allows for testing of more than 3 categories

Question 2

What are the different functions of the Chi square tests?

(3 different ways)

Answer 2

Goodness of fit

Homogeneity

Independence

Question 3

How can χ2 tests be used for goodness of fit

Answer 3

Goodness of a fit tests whether the observed values match up with the expected values.

They test a hypothesis about the distribution (model) of a qualitative variable in a population

I.e. do eye colours follow the following distribution? Brown 45%, blue 27%, hazel 18%, green 10%, etc and tthen compare this to the observed value (these are the expected values)

Question 4

How can χ2 tests be used to test for homogeneity

Answer 4

Tests a hypothesis about the distribution of a qualitative variable in several populations

Question 5

How can χ2 tests be used to test for independence

Answer 5

Tests a hypothesis about the relationship between two qualitative variables in a population - i.e. whether they are independent or not (is there an association between the two)

I.e. is there an association between parent eye colour (qualitative) and child eye colour (qualitative)

Question 6

How is the test statistics for all of the tests above calculated

Answer 6

χ2 (test statistic) = sum of ( [observed frequency - expected frequency] ^2 / expected frequency )

Question 7

What are the hypotheses for the chi square test for goodness of fit

Answer 7

H0 - model fits data/expected frequencies.

H1 - Model doesn’t fit data

Question 8

What are the assumptions involved in chi square tests for goodness of fit

Answer 8

None of the expected categories have a value of 0, and no more than 20% of the expected values are less than 5 (Cochran’s rule)

Question 9

What is Cochran’s rule

Answer 9

No more than 20% of the expected values are less than 5 - in other words, we want at least 80% of the results with an expected value of greater than 5

Question 10

How do we calculate the number of degrees of freedom from a chi square test

Answer 10

n-1

n = number of categories)

Question 11

How do we find the p-value from a chi square test

Answer 11

We use χ2 (n-1) curve to find upper tail area, n = number of categories

Question 12

How would we use chi square test to test for independence between two variables

Answer 12

We typically represent the data between two qual variables and two qual variables in a contingency table before putting it into a mosaic plot.

Question 13

What is the code for the chi square test

Answer 13

chisq.test(dataset)

Question 14

What are the hypotheses involved with the chi square test for independence

Answer 14

H: H0 - variables are independent

H1 - variables aren’t independent

Question 15

What are the assumptions involved with the chi square test for independence

Answer 15

Expected categories - none are empty, and no more than 20% are <5 (Cochran’s rule) - follows same assumptions as the chi square test for goodness of fit

Question 16

How do we calculate the test statistic from chi square test for independence

Answer 16

χ2 = sum of ([Observed frequency - expected frequency]) ^2 / expected frequency

Question 17

How do we calculate p - values for chi square test for independence

Answer 17

We use the χ2 curve to find upper tail area, however, this has a limit of degrees of freedom (df) of (m-1) (n-1), where m = number of categories from variable 1, and n = number of categories from variable 2

Question 18

How do we calculate degrees of freedom for chi square test for independence

Answer 18

We can use (m-1)(n-1) = df, where m = number of categories from variable 1 (i.e. a row), and n = number of categories from variable 2 (i.e. a column)

Question 19

What happens when df = 1, or there is a 2 x 2 contingency table

Answer 19

When this occurs, R automatically applies the Yates continuity correction onto the p - value obtained.

Question 20

What does Yates continuity correction do

Answer 20

It aims to make the test more conservative in terms of p values especially for small sample sizes or small number of categories as it could result in an increase in bias. This ultimately helps reduce the number of type 1 errors (false positives)

Question 21

How can we turn off Yates continuity correction

Answer 21

We can say correction = FALSE

Question 22

So what are the differences in the HATPC process for the use of chi square test for independence vs goodness of fit

Answer 22

Only differences are in the hypotheses, and the way the degrees of freedom are calculated

Question 23

Is it possible to use T tests to test the significance of a linear slope

Answer 23

Yes, it is possible, and it is often used

Question 24

What are the hypotheses typically involved in testing for the significance of the slope

Answer 24

H0 = no significant linear trend

H1 = significant linear trend

(normally)

Question 25

What are the assumptions involved in testing for the significance of a line)

Answer 25

Residuals need to be independent

Residuals should follow a normal distribution

Residuals should have a constant variance

Relationship between dependent and independent variable should look linear

Question 26

How can we check for homoscedasticity

Answer 26

Check residual plot for no observable pattern

Question 27

How can we check for residuals following normal distribution

Answer 27

Use a QQ plot and also a SHapiro Wilk test

Question 28

How can we check that residuals are independent

Answer 28

Check residual plot

Question 29

How can we check that there is a linear relationship between dependent and independent variables

Answer 29

Check the scatter plot for a linear relationship

Question 30

What is the test statistic for testing for significant linear relationship

Answer 30

T = (OV - EV) / SE, with n-2 degrees of freedom

Question 31

How is p - value obtained for linear relationship

Answer 31

We do the test statistic with n-2 degrees of freedom, and find the tail areas

We can also find it by doing the summary() output to give us the values that we want