Question

Consider the following statistical methods

(i) two-independent sample t-test (ii) paired t-test

(iii) chi-square goodness-of-fit test (iv) chi-square test of independence (v) t-test of a regression coefficient

(vi) F−test for multiple regression (vii) one-way ANOVA

(viii) two-way ANOVA

Compare and contrast three of the methods (your choice). Some questions to address: what type of data is used for each method? How are the methods similar? How are they different? Are there circumstances in which multiple methods can be used to arrive at the same conclusions?

Answer 1

The methods are: (i) two-independent sample t-test (ii) paired t-test and  (iv) chi-square test of independence

• Purpose :  (i) two-independent sample t-test - This test is used for comparing means of two independent normally distributed populations.  

(ii) paired t-test and  - This test is used for comparing means of two dependent populations which shoul also be normally distributed.

(iv) chi-square test of independence - This test is used for checking if there is any assoiation between two groups (characteristics) of a population.

• Type of data :  (i) two-independent sample t-test - Each of the  samples should be chosen randomly from the populations. The sample sizes must be large (>30) or the populations should be randomly distributed. The samples should be independent of each other.

(ii) paired t-test and  - The sample should contain two variables depending on each other. The sample sizes must be large (>30) or the populations should be randomly distributed, with unknown population variance.

   (iv) chi-square test of independence - The populations should consists of mutually exclusive classes.The classification may be with respect to either an atribute or a variable. In case of a continuous variable, the classification will necessarily be artificial , being achieved by dividing the whole range of the variable into k arbitrarily defined intervals . The samples should be drawn independently from each of the populations.

• Similarity : All of the methods assume randomness and of the samples, independent sample t and paired t both requires assumption of normality but chi sqaure does not. Again independent sample t and chi sqaure both assume indepennce of the samples, but paired t doesn't . Independent sample t and paired t both cpmpares means but chi sqare compares association.

• Difference :     Independent sample t calculates two sample measues (like mean, s.d) separately and then it is used to calculate the test statistic. Paired t calculates the differences of each data points for the samples and then calculate the measures of those differences (mean,s.d) to compute the test statistics. This Chi squared test uses observed frequencies to compute the expected frequencies to calculate the test statistics.

• There are no such circumstances where theese three methods can be used to arrive at the same conclusions .