The Different Types of Chi-square Tests and Their Purpose
Chi-square tests
1. INTRODUCTION
1.1 χ2 distribution and its properties
A chi-square (χ2) distribution is a set of density curves with each curve described by its degree of freedom (df). The distribution have the following properties:
- Area under the curve = 1
- All χ2 values are positive i.e. the curve begins from 0 (except for df=1) increases to a peak and decreases towards 0 as its asymptote
- The curve is skewed to the right, and as the degree of freedom increases, the distribution approaches that of a normal distribution Fig. 1 Graph of χ2 distribution with differing degrees of freedom
Each χ2 value is computed by the formula: χ2 = Σ (O-E)2 E where O = observed counts from the sample Equation 1 and E= expected counts based on the hypothesized distribution
1.2 Types of χ2 tests and their purpose
For a single population, to determine if the observed distribution in the population conforms to a specific known distribution or a previously studied distribution, the χ2 test for goodness-of-fit can be used.
An example of this usage include: Mendel’s genetic model predicts that the phenotypic distribution of two phenotypes, each phenotype having a dominant and recessive allele, will follow the ratio of 9:3:3:1. A study done to confirm this makes use of χ2 test for goodness-of-fit to determine if the observed population fits into the theoretical model. We will discuss this example in detail in the next section.
To compare the distribution of two populations, the χ2 test for homogeneity of population can be used. The data in this case can be represented in a two way table with the different populations in the rows and the distribution data based on certain categorical variable in columns. To test if the distribution of categorical variable is the same across several populations, the χ2 test for homogeneity of population is used.
An example of this can be to find out if the proportions of teachers with PhD teaching
1. INTRODUCTION
1.1 χ2 distribution and its properties
A chi-square (χ2) distribution is a set of density curves with each curve described by its degree of freedom (df). The distribution have the following properties:
- Area under the curve = 1
- All χ2 values are positive i.e. the curve begins from 0 (except for df=1) increases to a peak and decreases towards 0 as its asymptote
- The curve is skewed to the right, and as the degree of freedom increases, the distribution approaches that of a normal distribution Fig. 1 Graph of χ2 distribution with differing degrees of freedom
Each χ2 value is computed by the formula: χ2 = Σ (O-E)2 E where O = observed counts from the sample Equation 1 and E= expected counts based on the hypothesized distribution
1.2 Types of χ2 tests and their purpose
For a single population, to determine if the observed distribution in the population conforms to a specific known distribution or a previously studied distribution, the χ2 test for goodness-of-fit can be used.
An example of this usage include: Mendel’s genetic model predicts that the phenotypic distribution of two phenotypes, each phenotype having a dominant and recessive allele, will follow the ratio of 9:3:3:1. A study done to confirm this makes use of χ2 test for goodness-of-fit to determine if the observed population fits into the theoretical model. We will discuss this example in detail in the next section.
To compare the distribution of two populations, the χ2 test for homogeneity of population can be used. The data in this case can be represented in a two way table with the different populations in the rows and the distribution data based on certain categorical variable in columns. To test if the distribution of categorical variable is the same across several populations, the χ2 test for homogeneity of population is used.
An example of this can be to find out if the proportions of teachers with PhD teaching
References: Neil A. Campbell and Jane B. Reece. Biology. 7th Edition. Pearson publishing. 2005. Daniel S