Chi-Square Statistic

What is a Chi-Square Statistic?

Prerequisites

Knowledge about Proportions
Knowledge about Frequency distributions

Parametric

Hypothesis statistical tests are designed to test specific population parameters. Because these tests require assumptions about parameters, they are called parametric tests.
For example, t-tests are used to assess hypotheses about a population mean (µ) or mean difference (µ1 – µ2). These tests make assumptions about population parameters.
Parametric tests require a numerical score for each individual in the sample. The scores then are added, squared, averaged, and otherwise manipulated using basic arithmetic.
In terms of measurement scales, parametric tests require data from an interval or a ratio scale.

Non-parametric Statistical Test

Situations that do not conform to the requirements of parametric tests. In these situations, it may not be appropriate to use a parametric test. Remember that when the assumptions of a test are violated, the test may lead to an erroneous interpretation of the data. Fortunately, there are several hypothesis testing techniques that provide alternatives to parametric tests.
These alternatives are called non-parametric tests and introduce two commonly used examples of non-parametric tests.
Both tests are based on a statistic known as chi-square and both tests use sample data to evaluate hypotheses about the proportions or relationships that exist within populations.
Two chi-square tests, like most non-parametric tests, do not state hypotheses in terms of a specific parameter and they make few (if any) assumptions about the population distribution. For the latter reason, non-parametric tests sometimes are called distribution-free test

Difference between parametric & non-parametric test

One of the most obvious differences between parametric and non-parametric tests is the type of data they use.
All of the parametric tests require numerical scores.
In non-parametric tests, data is classified into categories such as High, Medium or Low
These classifications involve measurement on nominal or ordinal scales, and they do not produce numerical values.
Data for many non-parametric tests are simply frequencies. Example: Number of students scoring grades more than 9 and the number of students scoring below 9 grade in a class of n= 45 students.
Sometimes considering the nature of data it is better to obtain category measurements. For example, it is easier to classify scores as high, medium, or low in the place of a numerical score measuring for individuals.
In some cases, the Parametric test is preferred because it is more likely to detect a real difference or a real relationship.
Changing a parametric test into a non-parametric test usually involves transforming the data from numerical scores to non-numerical categories.

Example:

High, medium, and low scores.
Cold, Hot coffee
High-income, Low income, Economical weak sections
Extra small, Small, Medium, Large, and Extra Large –sizes
t-tests and ANOVA assumes that data come from normal distributions and are homogeneous. If data do not satisfy these assumptions, it may be better to transform the scores into categories and use a non-parametric test to evaluate the data.
Variance is a major component of the standard error in the denominator of t-statistics and the error term in the denominator of F-ratios.
Large variance can reduce the likelihood of significant differences.
Converting the scores into categories eliminates the variance. Example: Scores fit into three categories (high, medium, and low) no matter how much variance the original scores have.

Chi-square test

The chi-square test for goodness of fit uses sample data to test hypotheses about the shape or proportions of population distribution. The test determines how well the obtained sample proportions fit the population proportions specified by the null hypothesis.

Assumptions of Chi-Square Tests

Independence of Observations:

This is not to be confused with the concept of independence between variables, One consequence of independent observations is that each observed frequency is generated by a different individual.
A chi-square test would be inappropriate if a person could produce responses that can be classified in more than one category or contribute more than one frequency count to a single category.
Size of Expected Frequencies A chi-square test should not be performed when the expected frequency of any cell is less than 5. The chi-square statistic can be distorted when f e is very small.

Null Hypothesis :

The Null Hypothesis specifies the proportion (or percentage) of the population in each category.

Hypothesis: No Preference, Equal Proportions

The null hypothesis often states that there is no preference among the different categories. In this case, H0 states that the population is divided equally among the categories.

Example:

The hypothesis states that 50% of all Engineers who passed out of college are Boys and 50% are girls.

H0 Hypothesis: No Difference from a Known Population

H1 Hypothesis: Passed-out Engineers are not equally divided into Boys & Girls.

Null hypothesis: Chi-square test

Proportions for one population are not different from the Know proportions of the existing population.

Example:

There are 70% % of licensed Truck drivers and the remaining 30% are driving without a license in a state.
Second Population of other state drivers earning data was collected and categorized as High & Low

H₀ : Licence proportions of the drivers = Drivers Earning proportions

H1 : Drivers’ Earning is disproportionately high for Drivers (with a license) and disproportionately low for drivers (without a license).

Observed frequency

The observed frequency is the number of individuals from the sample who are classified in a particular category. Each individual is counted in one and only one category.

Expected frequency

The expected frequency for each category is the frequency value that is predicted from the proportions in the null hypothesis and the sample size (n). The expected frequencies define an ideal, hypothetical sample distribution that would be obtained if the sample proportions were in perfect agreement with the proportions specified in the null hypothesis

Goodness-of-Fit Test

The general purpose of any hypothesis test is to determine whether the sample data support or disprove a hypothesis about the population.
In the chi-square test for goodness of fit, the sample is expressed as a set of observed frequencies ( fo values), and the null hypothesis is used to generate a set of expected frequencies ( fe values). The chi-square statistic simply measures how well the data ( fo ) fit the hypothesis ( fe ). The symbol for the chi-square statistic is χ2. The formula for the chi-square statistic is

The formula for chi-square involves adding squared values, so you can never obtain a negative value. Thus, all chi-square values are zero or larger.
When H₀ is true, data ( f_o values) is close to the hypothesis ( f_e values). Chi-square values will be small when H₀ is true.
The above two factors suggest that the typical chi-square distribution will be positively skewed.
Each specific chi-square distribution is identified by degrees of freedom (df).

df = C – 1, where C is the number of categories

Chi-square values tend to get larger (shift to the right) as the number of categories and degrees of freedom increase.

Critical Region for a Chi-Square Test

When the null hypothesis is true and df = 5, only 5% (.05) of the chi-square values are greater than 11.07, and only 1% (.01) are greater than 15.09.

Thus, with df = 3, any chi-square value greater than 11.07 has a probability of p < .05, and any value greater than 15.09 has a probability of p < .01

Example:

The choice was given to select one habit out of four options (Walking, Brisk walking, Running and gym) to maintain a healthy lifestyle. n = 50 participants have given their feedback. The following data indicate the number of votes in each category given by people.

The question for the hypothesis test is whether there are any preferences among the four possible options. Are any of the options selected more (or less) often than would be expected simply by chance?

Solution:

n = 132

Hypotheses H₀ : there is no preference for any option
Four possible options are selected equally and population distribution has the following proportions:

H1: In the general population, one or more of the options is preferred over the others. use α = .05.

Degree of freedom:

No of choice C = 4 , df = C -1 =3
df = 3 & α = 0.05. χ2 = 7.81

Now calculate the chi-square statistic

Expected frequency = = 132 * 0.25 = 33

χ²calculated is less than the critical χ² (3, .05 ) = 7.81

State a decision and a conclusion

The obtained chi-square value is not in the critical region. Therefore, H0 is accepted.
There are no significant differences among the four options, it is because of chance.

Results of Chi-Square

The participants showed no significant preferences among the four options for a healthy life,
χ2 (3, n = 132) =7.81, p < .05

Independent Variable

Two variables are independent when there is no consistent, predictable relationship between them. In this case, the frequency distribution for one variable is not related to (or dependent on) the categories of the second variable. As a result, when two variables are independent, the frequency distribution for one variable will have the same shape (same proportions) for all categories of the second variable.

Chi-Square Test for Independence

The chi-square test for independence uses the frequency data from a sample to evaluate the relationship between two variables in the population.
Each individual in the sample is classified on both of the two variables, creating a two-dimensional frequency distribution matrix.
The frequency distribution for the sample is then used to test hypotheses about the corresponding frequency distribution in the population.

Null Hypothesis

H₀1 : For the general population, there is no relationship between the options selected and the personality category.
H₀2 : In the population of Males, the proportions in the distribution of Options for good health are not different from the proportions in the distribution of options preferences for females. The two distributions have the same shape (same proportions)

f_c is the frequency total for the column (column total),

f_r is the frequency total for the row (row total)

n is the number of individuals in the entire sample

df = (R – 1)(C – 1) = (2-1) (4-1) = 3

calculate fa

Calculate fe

Calcuate fa -fe

Calculate (fa -fe)^2

Calculate (fa -fe)^2/ fe

Calculate the sum of (fa -fe)^2/ fe = 10.39

χ2 = 10.39

chi-square from table =

{at df = 3 and α = 0.05.}

χ2 = 7.81

χ2 calculated (10.39 ) is in critical region (> 7.81) and therefor H0 is rejected.

Proportions in the distribution of Options for good health are different from the proportions in the distribution of options preferences for Females.

Chi-Square Tests Effect Size measurement

Hypothesis tests, like the chi-square test for goodness of fit or for independence, evaluate the statistical significance of the results.
The test determines whether it is likely that the patterns or relationships observed in the sample data could have occurred without any corresponding patterns or relationships in the population.
Tests of significance are influenced not only by the size or strength of the treatment effects but also by the size of the samples.
A small effect can be statistically significant if it is observed in a very large sample. Because a significant effect does not necessarily mean a large effect, it is generally recommended that the outcome of a hypothesis test be accompanied by a measure of the effect size.
Cohen (1992) introduced a statistic called w that provides a measure of effect size for either of the chi-square tests. The formula for Cohen’s w is very similar to the chi-square formula but uses proportions instead of frequencies.

Chi-Square Statistic

Magnitude of w

Values near 0.10 indicate a small effect
Values 0.30 a medium effect
and 0.50 a large effect

Role of Sample Size

The formula for computing w does not contain any reference to the sample size.
w is calculated using only the sample proportions and the proportions from the null hypothesis.
The size of the sample has no influence on the magnitude of w. This is one of the basic characteristics of all measures of effect size.
The number of scores in the sample has little or no influence on effect size.

Chi-square Vs Effect size (W)

chi-square statistic and effect size as measured by w are intended for different purposes and are affected by different factors.
The formula for w which is under the square root, can be obtained by dividing the formula for chi-square by n. Dividing by the sample size converts each of the frequencies (observed and expected) into a proportion, which produces the formula for w

Phi-Coefficient

phi-coefficient as a measure of correlation for data consisting of two dichotomous variables (both variables have exactly two values).
This same situation exists when the data for a chi-square test for independence form a 2 × 2 matrix (again, each variable has exactly two values).
It is possible to compute the correlation phi (ϕ) in addition to the chi-square hypothesis test for the 2 x2 matrix data.
phi is a correlation, it measures the strength of the relationship, rather than the significance, and thus provides a measure of effect size.
The value for the phi-coefficient can be computed directly from chi-square by the following formula:

Cramér’s V

When the chi-square test involves a matrix larger than 2 × 2, a modification of the phi-coefficient, known as Cramér’s V, can be used to measure effect size.

The formula for Cramér’s V is identical to the formula for the phi-coefficient, except for the addition of df* in the denominator.
The df* value is not the same as the degrees of freedom for the chi-square test, but it is related.
df* is the smaller of either (R – 1) or (C – 1).

Median Test for Independent Samples

The median test provides a non-parametric alternative to the independent-measures t-test (or ANOVA) to determine whether there are significant differences among two or more independent samples.
The null hypothesis for the median test states that the different samples come from populations that share a common median (no differences).
The alternative hypothesis states that the samples come from populations that are different and do not share a common median.
The logic behind the median test is that whenever several different samples are selected from the same population distribution, roughly half of the scores in each sample should be above the population median and roughly half should be below.
That is, all the separate samples should be distributed around the same median. On the other hand, if the samples come from populations with different medians, then the scores in some samples will be consistently higher and the scores in other samples will be consistently lower
The first step in conducting the median test is to combine all the scores from the separate samples and then find the median for the combined group.
A matrix is constructed with a column for each of the separate samples and two rows: one for individuals above the median and one for individuals below the median.
Finally, for each sample, count how many individuals scored above the combined median and how many scored below. These values are the observed frequencies that are entered into the matrix.
The frequency distribution matrix is evaluated using a chi-square test for independence. The expected frequencies and a value for chi-square are compute
A significant value for chi-square indicates that the discrepancy between the individual sample distributions is greater than would be expected by chance.
The median test does not directly compare the median from one sample with the median from another. Thus, the median test is not a test for significant differences between medians. Instead, this test compares the distribution of scores for one sample versus the distribution for another sample.
If the samples are distributed evenly around a common point (the group median), the test will conclude that there is no significant difference. On the other hand, finding a significant difference simply indicates that the samples are not distributed evenly around the common median.
The best interpretation of a significant result is that there is a difference in the distributions of the samples.

What is a Chi-Square Statistic?

Prerequisites

Parametric

Non-parametric Statistical Test

Difference between parametric & non-parametric test

Example:

Chi-square test

Assumptions of Chi-Square Tests

Null Hypothesis :

Example:

Null hypothesis: Chi-square test

Example:

Observed frequency

Expected frequency

Goodness-of-Fit Test

Critical Region for a Chi-Square Test

Example:

Solution:

Degree of freedom:

Now calculate the chi-square statistic

State a decision and a conclusion

Results of Chi-Square

Independent Variable

Chi-Square Test for Independence

Null Hypothesis

Chi-Square Tests Effect Size measurement

Magnitude of w

Role of Sample Size

Chi-square Vs Effect size (W)

Phi-Coefficient

Cramér’s V

Median Test for Independent Samples

Also read :

Related Posts