Parametric and Non-Parametric Tests, Goodness of Fit, Independence

This assignment solution provides a comprehensive overview of parametric and non-parametric statistical tests. It explains the assumptions underlying parametric tests, such as normality and homogeneity of variance, and contrasts them with non-parametric tests, which are used when these assumptions are not met. The document details the application of goodness-of-fit tests, such as the Chi-square test, to assess the similarity between observed and expected values. Furthermore, it describes the test of independence, specifically the Chi-square test of independence, used to examine the association between two categorical variables, using contingency tables for analysis. Examples are provided to illustrate the practical use of these statistical methods, along with relevant references.

Solution:
Parametric tests make certain assumption about the parameter a sample has been picked up.
Parametric tests are used only when the distribution is normal. Parametric tests also assume
homogeneity of within group variances. For parametric tests, the variables should be at interval
level. If the results of parametric test are to be reliable, one must satisfy all the condition
mentioned above. For example, a t-test is reliable only when it satisfies the conditions of normal
distribution and equal variances are met. Parametric tests do not work well when the distribution
is non-normal because of outliers, skewness, kurtosis, etc. In such circumstances, non-parametric
tests are used. Non-parametric tests do not make any assumptions about the shape of the
distribution. Parametric test generally use mean while non-parametric tests use median.
Parametric tests are more robust than non-parametric tests because they have more power – in
other words, with parametric test, one is more likely to reject a null hypothesis as compared to
non-parametric test.
Goodness of fit test examines whether observed value is equal to expected values/fitted values.
In other words, it tells us how similar your data is to the actual population. Some of the
commonly used goodness-of-fit test are Chi-square, Kolmogorov-Smirnov, etc. Example: a
company sells jars of assorted chocolates. The company manufactures six different type of
chocolates. Each jar, the company claims, has equal number of chocolates of each color. In this
case, we would like to test whether company’s claim of each jar having equal amount of
chocolates of each colour is correct. We can make use of Chi-square goodness of fit test to
examine this.
Test of independence is used to test the association between two variables taken from the same
population. Chi-square test of independence is used to test the association between two
categorical variables. For example – in a health survey, suppose we aim to examine whether the
size of the children at birth (small, medium, large) is related with type of antenatal care (full,
partial). In such a case one has to use Chi-square test of independence. In this test, the frequency
of one nominal variables is compared with another nominal variable to test whether there exists
an association between those two variables. Contingency table is used for this purpose. The
values for one variable are kept in row and for another is kept in column.

A two-way table is best for summarizing the counts/frequency of two categorical variables. Once
can examine relationship between two categorical variables using such tables. An example of a
two-way table is given below, which seeks to uncover the relationship between voting pattern
and religion.
Table1: Proportion of votes by religion.
Religion Republican Democrat
Christian 0.25 0.30
Muslim 0.22 0.15
Jew 0.35 0.40
Others 0.18 0.15
Since we have two categorical variables year and companies and we want to establish whether an
association exist between current shares and 1990 shares, we will apply Chi-square test of
independent to determine whether the current market shares differ from those of 1990.
When the aim is to test the association between two categorical variables, the Chi-square
test of independence is used.
H0: There is no difference between current market shares and 1990 market shares.
H1: There is a difference between current market shares and 1990 market shares.
References:
1. Sheskin, D. J. (2004). Handbook of parametric and nonparametric statistical procedures
(3rd ed.). Boca Raton, FL, : Chapman & Hall/CRC.
2. Gujarati, D. N., & Porter, D. C. (2009). Basic econometrics. Boston, Mass: McGraw-Hill.
3. Witte, R. S., & Witte, J. S. (2017). Statistics. Hoboken, NJ: Wiley.