Chi-Square Test Presentation__Method.pdf
- 2. Introduction Chi-square is one of the most commonly used non-parametric test Introduced by Karl Pearson as a test of significance in 1990 Denoted by the Greek sign χ2 It is a useful measure of comparing experimentally observed result with experimentally theoretical result or based on hypothesis The chi-squared distribution with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random samples It is determined by the degrees of freedom It can be applied on categorical or qualitative data using a contingency table Used to evaluate unpaired/unrelated samples and proportions
- 3. Chi-Square It is a mathematical expression, representing the ratio between experimentally obtained result (O) and the theoretically expected result (E) based on certain hypothesis It used data in the form of frequencies (i.e., the number of occurrence of an event) Calculated by dividing the square of the overall deviation in the observed and expected frequencies by the expected frequency
- 4. If there is no difference between the actual and observed frequency, the value of chi-square is zero If there is a difference then the value of test will be other than zero Differences may be due to sampling fluctuations FORMULA: CHI SQUARE
- 5. Contingency Table A type of table in a matrix format that displays the multivariate frequency distribution of the variables It provides a basic picture of interrelation between two variables The values depend on the number of classes
- 6. Degrees of Freedom It is the number of independent pieces of information which are free to vary, that go into the estimate of a parameter In a contingency table, the degree of freedom is calculated in a different manner as df = (R-1) (C-1) where: R = no. of rows in a table C = no. of columns in a table
- 7. Chi-Square Distribution The sampling distribution of the chi-square statistic is not a Normal distribution. It is a right-skewed distribution that allows only positive values because X2 can never be negative When the expected counts are all at least 5, the sampling distribution of the X2 statistic is close to a chi-square distribution with df equals the number of categories minus 1.
- 8. Characteristics of Chi Square It is based on frequencies and not on the parameters like mean and standard deviation Used for testing difference between the entire set of the expected and the observed frequency Used for testing the hypothesis and is not useful for estimation It is an important non-parametric test as no rigid assumptions are necessary in regard to the type of population, no need of parameter values and relatively less mathematical details are involved
- 9. Assumptions for the validity of Chi-Square Test All observations should be independent No individual item should be included twice. The total number of observation should be large. The chi-square test should not be used if n>50 For comparison purpose, the data must be in original units If the theoretical frequencies is <5, then we pool it with either preceding or succeeding frequency, so that the resulting sum is >5
- 10. Limitations It does not give us much information about the strength of the relationship. It only conveys the existence on non-existence of relationships between the variables It is sensitive to sample size It is also sensitive to small expected frequencies.
- 11. STEPS Identify the problem Make a contingency table and note the observed frequency (O) in each classes of one event, row wise i.e., horizontally. And then the numbers in each group of the other event. column wise, i.e., vertically Set up the Ho; According to Null Hypothesis, no association exists between attributes. This needs setting up of Ha Calculate the expected frequencies (E) Find the difference between observed and expected frequency in each cell (O-E) Calculate the chi-square value by applying the formula. The value ranges from zero to infinite. 1. 2. 3. 4. 5. 6.
- 12. Uses of Chi Square Test Goodness of fit - It measures how much the observed or actual frequency differ from the expected/predicted frequency. Test of Homogenity - Used to determine whether frequency counts are distributed identically across different samples Test of Independence - Used to explain that variables are how much attached with each other. 1. 2. 3.
- 13. Chi-Square Test of Goodness-of-Fit Chi-Square Test of Independence TYPES Number of Variables: One Purpose of Test: Determines if sample date matches a population Degrees of Freedom: K-1 Number of Variables: Two Purpose of Test: Compares two set of data to see if there is a relationship Degrees of Freedom: (r-1) (c-1)
- 14. Chi Square Goodness of Fit Test A Ho and Ha established and a significance level is selected for rejection of Ho. A random sample of observations is drawn from a relevant statistical population. A set of expected frequencies is derived under the assumption that the Ho is true The observed frequencies compared with the expected frequencies The calculated value of Chi-Square goodness of fit test is compared with the table value. If the calculated value of chi-square goodness of fit is greater than the table value, we will reject the null hypothesis and conclude that there is a significant difference between the observed and the expected frequency. 1. 2. 3. 4. 5.
- 15. Steps in Testing Goodness of Fit A null and alternative hypothesis is established and a significance level is selected for rejection of null hypothesis. A random sample of observations is drawn from a relevant statistical population. A set of expected frequencies is derived under the assumption that the null hypothesis is true. The observed frequencies compared with the expected frequencies The calculated value of Chi-Square goodness of fit test is compared with the table value. If the calculated value of the Chi-Square goodness of fit test is greater than the table value, we will reject the null hypothesis and conclude that there is a significant difference between the observed and the expected frequency. 1. 2. 3. 4. 5.
- 16. Conditions for Performing a Chi-Square for Goodness of Fit Random: The data come a well-designed random sample or from a randomized experiment 10%: When sampling without replacement, check that n ≤ (1/10)N. Large Counts: All expected counts are greater than 5 Before we start using the chi-square goodness-of-fit test, we have two important cautions to offer. •The chi-square test statistic compares observed and expected counts. Don’t try to perform calculations with the observed and expected proportions in each category. •When checking the Large Sample Size condition, be sure to examine the expected counts, not the observed counts.
- 17. Mars, Incorporated makes milk chocolate candies. Here’s what the company’s Consumer Affairs Department says about the color distribution of its M&M’S® Milk Chocolate Candies: On average, the new mix of colors of M&M’S ® Milk Chocolate Candies will contain 13 percent of each of browns and reds, 14 percent yellows, 16 percent greens, 20 percent oranges and 24 percent blues. The one-way table summarizes the data from a sample bag of M&M’S ® Milk Chocolate Candies. In general, one-way tables display the distribution of a categorical variable for the individuals in a sample. Sample Problem:
- 18. Stating the Hypothesis H0: The company’s stated color distribution for M&M’S ® Milk Chocolate Candies is correct. Ha: The company’s stated color distribution for M&M’S ® Milk Chocolate Candies is not correct. We can also write the hypotheses in symbols as: H0: pblue= 0.24, porange= 0.20, pgreen= 0.16, pyellow= 0.14, pred= 0.13, pbrown= 0.13, Ha: At least one of the pi’s is incorrect
- 19. Finding the expected counts for the color categories:
- 20. To compute for the Chi-Square Statistic:
- 21. The Chi-Square Distributions and P-values Since our P-value is between 0.05 and 0.10, it is greater than α = 0.05. Therefore, we fail to reject H0. We don’t have sufficient evidence to conclude that the company’s claimed color distribution is incorrect.