Statistics Ch. 14 Notes

LEARNING +ROM CATEGORICAL DATA SECTION 14.1 - Chi-Squared Tests For Univariate Categorical Data • Univariate categorical data sets arise in a variety of settings • Univariate categorical data are usually summarized in a one- way frequency table. Notation - k = number of categories of a categorical variable - p1 = population proportion for category 1 - p2 = population proportion for category 2 - pk = population proportion for category k Note: p1 + p2 + ... + pk = 1 • The hypotheses tested have the form - Ho: p1 = hypothesized proportion for category 1 - Ho: p2 = hypothesized proportion for category 2 - Ho: pk = hypothesized proportion for category k • Ha: Ho is not true. At least one of the population category proportions differs from the corresponding hypothesized value • The expected counts for a particular null hypothesis are calculated using the hypothesized proportions from the null hypothesis • A null hypothesis can be tested by first selecting a random sample of size n and then classifying each response into one of the k possible categories • To decide whether the sample data are compatible with the null hypothesis, you compare each observed count (the observed category frequency) to an expected count • The goodness-of-fit statistic, denoted by X^2, is a quantitative measure of the extent to which the observed counts differ from those expected when Ho is true The goodness-of-fit statistic, X², results from calculating (observed count - expected count)² / expected count for each category, where for a sample of size n, expected count = n (hypothesized value of corresponding population proportion) • The X² statistic is the sum of these quantities for all k categories: X² = Σ (observed count - expected count)² / expected count all categories • The value of the X² statistic reflects the size of the differences between observed and expected counts • When the differences are big, the value of X² tends to be large, which suggests Ho should be rejected • When the null hypothesis is true and the sample size is large, the behavior of X² is described approximately by a chi-square distribution • A chi-square distribution curve has no area associated with negative values and is not symmetric, with a longer tail on the right Chi-Square Curves A Chi-Square Upper-Tail Area [Hand-drawn graph showing two chi-square distribution curves] [Hand-drawn graph showing a chi-square distribution curve with the right tail shaded] CHI-SQUARE GOODNESS-OF-FIT TEST • When a null hypothesis that specifies k category proportions is true, the X² goodness-of-fit statistic has approximately a chi-square distribution with df = k - 1, as long as none of the expected cell counts are too small • When expected counts are small, and especially when an expected count is less than 1, the value of (observed count - expected count)² / expected count can be inflated because it involves dividing by a small number • The use of the chi-square distribution is appropriate when the sample size is large enough for every expected count to be at least 5 CHI-SQUARE GOODNESS-OF-FIT TEST • Appropriate when the following conditions are met: - Observed counts are based on a random sample or a sample that is representative of the population - The sample size is large. The sample size is large enough for the chi-square goodness-of-fit test to be appropriate if every expected count is at least 5 • When these conditions are met, the following test statistic can be used: x² = Σ (observed count - expected count)² / expected count all categories • When the null hypothesis is true, the x² statistic has a chi-square distribution with df = k - 1, where k is the number of category proportions specified in the null hypothesis • Hypotheses - Ho: p1 = hypothesized proportion for category 1 - Ho: p2 = hypothesized proportion for category 2 - Ho: pk = hypothesized proportion for category k • Ha: Ho is not true. At least one of the population category proportions differs from the corresponding hypothesized value • Associated P-value - The P-value is the area to the right of x² under the chi-square curve with df = k - 1 SECTION 14.2 - Tests For Homogeneity and Independence in a Two- Way Table • Bivariate categorical data are usually summarized in a two-way frequency table - This is a table that consists of a row for each possible category of one of the variables and a column for each possible category of the other variable - There is a cell for each possible combination of category values - The number of times each particular combination occurs in the data set is entered in the corresponding cell of the table - These numbers are called observed cell counts • Marginal totals are obtained by adding the observed cell counts in each row and also in each column of the table The row and column marginal totals, along with the total of all observed cell counts in the table is called the grand total • Two-way tables are also used when data are collected to compare two or more populations or treatments on the basis of a single categorical variable COMPARING TWO OR MORE POPULATIONS OR TREATMENTS: A TEST OF HOMOGENEITY • When the purpose of a study is to compare two or more populations or treatments on the basis of a categorical variable, the question of interest is whether the category proportions are the same for all the populations or treatments • The test procedure uses a chi-square statistic to compare the observed counts to those that would be expected if there were no differences among proportions • To compare two or more populations or treatments on the basis of a categorical variable, calculate an expected cell count for each cell by selecting the corresponding row and column marginal totals and then using the formula - expected cell count = (row marginal total) (column marginal total) / grand total • These expected cell counts represent what is expected if there are no differences in the category proportions for the groups Chi-Square Test For Homogeneity • Appropriate when the following conditions are met: - Observed counts are from independently selected random samples, or subjects in an experiment are randomly assigned to treatment groups - The sample sizes are large. The sample sizes is large enough for the chi-square test for homogeneity to be appropriate if every expected count is at least 5. If some expected counts are less than 5, rows or columns of the table may be combined to achieve a table with satisfactory expected counts • When these conditions are met, the following test statistic can be used: - x² = Σ (observed count - expected count)² / expected count all cells • The expected cell counts are estimated from the sample data with - expected cell count = (row marginal total) (column total) / grand total When conditions above are met and the null hypothesis is true, the X² statistic has a chi-square distribution with df = (number of rows - 1)(number of columns - 1) • Hypotheses - Ho: The population (or treatment) category proportions are the same for all the populations (or treatments) - Ha: The population (or treatment) category proportions are not all the same for all of the populations (or treatments) • Associated P-value - The p-value associated with the calculated value of the test statistic is the area to the right of X² under the chi-square curve with df = (number of rows - 1)(number of columns - 1) Chi-Square Test For Independence • Appropriate when the following conditions are met: - Observed counts are from a random sample or a sample that is selected in a way that would result in it being representative - The sample size is large. The sample size is large enough for the chi-square test for independence to be appropriate if every expected counts are less than 5, rows or columns of the table can be combined to achieve a table with satisfactory expected counts • When these conditions are met, the following test statistic can be used: X² = Σ (observed count - expected count)² / expected count all cells • The expected cell counts are estimated from the sample data with expected cell count = (row marginal total)(column total) / grand total • When conditions above are met and the null hypothesis is true, the X² statistic has a chi-square distribution with df = (number of rows - 1)(number of columns - 1) • Hypotheses - Ho: The population (or treatment) category proportions are the same for all the populations (or treatments) - Ha: The population (or treatment) category proportions are not all the same for all of the populations (or treatments) • Associated P-value - The p-value associated with the calculated value of the test statistic is the area to the right of X² under the chi-square curve with df = (number of rows - 1)(number of columns - 1)