# Some Critical Information about SOME Statistical Tests and Measures of Correlation/Association

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1 Some Critical Information about SOME Statistical Tests and Measures of Correlation/Association This information is adapted from and draws heavily on: Sheskin, David J Handbook of Parametric and Nonparametric Statistical Procedures. Second Edition. Chapman & Hall/CRC, Boca Raton. 982 pp. I strongly recommend that you consult Sheskin for other statistical tests and measures of correlation. This is simply a short, incomplete description of some of the limitations and assumptions for commonly used tests and measures of association. My objective is to show the relationship between what kind of data you collect, how many samples or populations you include, and how the sample was selected (probabilistic or not) and the kinds of statistical tests you can use to analyze your data. Consult a statistician (preferable) or at least a statistics book before you use any statistical procedure. Sheskin provides much more detail about these and many other statistical tests and measures. In short, use this information with great care!!! I am not a statistician. Critical Terminology & Concepts You have had an introduction to the terminology and concepts in statistics. Remember the following points. They will be critical to making sense of this guide. There are four levels of measurement ratio, interval, ordinal and nominal. If you do not remember what these mean, refer to Nardi. Review Nardi if you cannot remember what the terms variance and standard deviation mean. Review the Bernard reading on sampling if you cannot remember what the terms normal distribution, skewness and kurtosis mean. Use must have ratio or interval data to conduct parametric statistical tests. Non-parametric tests are used for ordinal and nominal data, although the kinds of tests that you can perform on nominal data are very few. There are three measures of central tendency. The mean can only be calculated for ratio and interval data. The median can be calculated for ordinal data and the mode for nominal data. You can calculate the measures of central tendency for lower levels of data for higher levels of data e.g., you could calculate the mean, median and mode for ratio or interval data. A variable, a factor used for assignment to comparison groups, and a treatment are not the same thing, even though it is very common for people to use the same term for both. A variable, by definition, can assume different values. Scores on tests could range from 0 to 100 (ratio data). You couple ask people to rate their confidence in using statistics appropriately very low, low, moderate, high, very high (ordinal data). You could ask people their favorite food, pizza, hamburgers, sushi, spinach (nominal data). A factor used to assign people to comparison groups may have different values, but the point is that the researcher determines the values that a factor can assume in a study. For example, you might want to compare income for people based on educational level. You could assign people to four groups, less than high school, high school, some college, college degree. It would not matter if one participant had 1 year of high school and another had 3 years of high school both participants would be assigned to the less than high school group. While some authors will call this an independent variable, you must remember that you are not allowing this variable to assume the full range of actual values. You are simply using it to place participants in one of four comparison groups. A more Statistics 1

2 correct term, and the one I want you to use when referring to cross-sectional, longitudinal and case study designs, is the variable or condition used for assignment to comparison groups. For experiments, the treatment is not a variable by definition. The treatment is something you do to the participants. It is not allowed to vary at all. In ANOVA, we use the term factor rather than treatment for reasons unknown to me. Do not refer to the treatment as the independent variable. Call it the treatment or factor. Independent sample means that two or more samples are from different populations or experimental groups. You can only rarely treat pre- and post-test scores as independent samples because the pre-test score typically influences the post-test score. Dependent samples mean that the two samples are related in some way that will affect the outcome of the study. There are three common cases. (1) The most common use of the dependent sample tests are for comparing change between pre- and post-test scores. Since the pre-test score influences the post-test scores, the two samples are treated as dependent samples. Dependent samples also refer to cases where [2] each subject serves as a member of all treatment groups and of the control group as in a switching replications experimental design or [3] every subject is paired with a subject in each of the other test groups as in a matching samples cross-sectional or experimental design, in which case the pairing must be justified. You do this by identifying one or more variables other than the independent variable or factor used for determining comparison groups (like gender) that you believe are positively correlated with the dependent variable. You match subjects based on their similarity in regard to this (or these) variables. In experiments, you then randomly assign matched members to the comparison and treatment groups. In cross-sectional designs, the normal use occurs when the number of participants with the desired characteristics for the independent variable(s) is small preventing the researcher from using screening to eliminate potential participants. In order to try to prevent other characteristics known or suspected of association with the outcome variable, the researcher matches the subjects in the comparison group on these characteristics. TESTS OF CENTRAL TENDENCY I. Interval or Ratio Data - Parametric Tests Means Tests A. One sample 1. Single Sample z Test a) What it tests: Whether a sample of subjects or objects comes from a population does the sample mean equal the population mean? b) Limitations: You must know the standard deviation and mean of the population for the primary variable of interest. c) Assumptions: The sample represents the population. The sample was randomly selected. The population is normally distributed with regard to the variable of interest, which is usually the outcome or dependent variable, but can be an independent variable. 2. Single-Sample t Test a) What it tests: Whether a sample of subjects or objects represents a population does the sample mean equal the population mean for the variable of interest? b) Limitations: You must know the mean of the population for the variable of interest. Statistics 2

3 c) Assumptions: The sample represents the population. The sample was randomly selected. The population is normally distributed for the variable of interest. B. Two or more independent variables or factors 1. t Test for Two Independent Samples a) What it tests: Do two independent samples represent two different populations that have different mean values for the variable of interest (usually the outcome or dependent variable) b) Limitations: You can only compare two samples c) Assumptions: The samples are representative of the populations for the variable of interest. The samples were randomly selected. The samples are independent. Both populations are normally distributed for the variable of interest. The variances of the two populations are equal for the variable of interest. 2. Single Factor Between-Subjects or One Way Analysis of Variance (ANOVA) a) What it tests: In a group of any number of samples (three, five, ten), do at least two of the samples represent populations with different mean values for the variable of interest, usually the outcome or dependent variable? b) Additional procedures: This test does not tell you which of the means differed just that there was a difference between some of them. For planned comparisons you may use multiple t tests to determine which means differ. For unplanned tests you may use Fisher s LSD test to determine which means differ. c) Limitations: Only one variable or condition used for assignment to comparison groups in observational designs or factor in an experimental design. d) Assumptions: Samples are representative of the populations. The samples were selected randomly. The samples are independent. All of the populations are normally distributed for the variable of interest, usually the outcome or dependent variable. The variances of all of the populations are equal for the variable of interest. 3. Single Factor Between-Subjects Analysis of Covariance (ANCOVA) a) What it tests: It is a form of ANOVA. It allows you to use data about an independent variable that has a linear correlation with the dependent variable to (1) remove variability in the dependent variable and/or (2) adjust the mean scores of the different groups for any pre-existing differences in the dependent variable that were present prior to the administration of the experimental treatments. A commonly used covariate variable is a pretest score for the dependent variable which removes the effect of pre-test performance on the post-test score (e.g., this is a kind of surrogate measure for dependent samples which are described below). b) Limitations: Only one variable or condition used for assignment to comparison groups in observational designs or factor in an experimental design. c) Single factor ANCOVA is sometimes used for a design in which subjects are not randomly assigned to groups (quasi-experimental designs). This use is problematic! This includes in some cases using single factor ANCOVA for inferential designs (ex post facto studies where the group are based on something like sex, income or race). This is even more problematic. d) Assumptions: Samples are representative of the populations. All of the populations are normally distributed for the variable of interest. The variances of all of the populations are equal for the variable of interest. Statistics 3

4 C. Two or More Dependent Samples. 1. t Test for Two Dependent Samples a) What it tests: Do two dependent samples represent populations with different mean values for the variable of interest. b) Limitations: Only two samples (groups, populations) c) Samples are representative of the populations. Samples were randomly selected. Both populations are normally distributed for the variable of interest, which is often the outcome or dependent variable but can be an independent variable The variances of the two populations are equal for the variable of interest. 2. Single Factor Within-Subjects ANOVA a) What it tests: In a group of any number of dependent samples (three, five, ten), do at least two of the samples represent populations with different mean values? b) Additional procedures: This test does not tell you which of the means differed just that there was a difference between some of them. For planned comparisons you may use multiple t tests to determine which means differ. For unplanned tests you may use Fisher s LSD test to determine which means differ. c) Limitations: Only one variable or condition used for assignment to comparison groups in observational designs or factor in an experimental design. d) Assumptions: Samples are representative of the populations. Samples were randomly selected. All of the populations are normally distributed for the variable of interest. The variances of all of the populations are equal for the variable of interest. D. Two or More Samples and Two or More Independent Variables or Factors 1. Between-Subjects Factorial ANOVA a) What it tests: (1) Do at least two of the levels of each factor (A, B, C, etc.) represent populations with different mean values? (2) Is there an interaction between the factors? b) Additional procedures: Case 1 there were no significant main effects (no differences between factors) and there is no significant interaction. You can conduct any planned tests, but it is probably fruitless to conduct unplanned tests. Case 2 there were significant main effects (differences between factors), but there was no significant interaction. In this case you can treat the factors separately just ignore interaction. Use a single factor between-subjects ANOVA. Case 3 interaction is significant, whether or not there are any significant main effects (differences between factors). Use a single factor between-subjects ANOVA for all levels of one factor across only one level of the other factor. (E.g., hold one factor constant while you allow the other to vary.) c) Minor d) Assumptions: Samples are representative of the populations. Samples were randomly selected. Samples are independent. All of the populations are normally distributed for the variable of interest, usually the outcome variable. The variances of all of the populations are equal for the variable of interest. Statistics 4

5 II. Ordinal (Rank Order) Data - Nonparametric Tests Median Tests A. One Sample 1. Wilcoxon Signed-Ranks Test a) What it tests: Whether a sample of subjects or objects comes from a defined population does the sample median equal the population median? b) Limitations: You must know the median of the population. c) Assumptions: The sample is representative of the population. The sample was randomly selected. The population distribution is symmetrical for the variable of interest. B. Two or more independent samples 1. Mann-Whitney U Test a) What it tests: Do two independent samples represent two populations with different median values? b) Limitations: You can only compare two samples, no more. Do not use this test for proportions (percentages). c) Assumptions: The samples are representative of the populations for the variable of interest. The samples were randomly selected. The samples are independent. The original variable of interest that was measured was a continuous random variable (this assumption is often violated no idea if that s OK or not, but Sheskin does not seem to think it is a big deal). The distributions of the populations are identical in shape for the variable of interest. 2. Kruskal-Wallis One-Way Analysis of Variance by Ranks a) What it tests: In a group of any number of independent samples (three, five, ten), do at least two of the samples represent populations with different median values? b) Additional procedures: Like the parametric ANOVA, the Kruskal-Wallace test does not tell you which of the means differed. You must perform pairwise comparisons to determine where the differences lie. See a good statistics book about how to do this. You can use the Mann-Whitney U test but there are certain conditions that must be met for this procedure to work. Again consult a statistics book. c) Limitations: Only one variable or condition used for assignment to comparison groups in observational designs or factor in an experimental design. d) Assumptions: Samples are randomly selected. Samples are representative of the populations for the variable of interest. Samples are independent of one another. The original variable that was measured was a continuous random variable (this assumption is often violated no idea if that s OK or not, but Sheskin does not seem to think it is a big deal). The distributions of the populations are identical in shape for the variable of interest. C. Two or more dependent samples 1. Wilcoxon Matched Pairs Signed Ranks Test a) What it tests: Do two dependent samples represent two different populations? b) Limitations: Only two samples, no more. You must have two scores to compare for this test because it is based on the difference between the two. These can Statistics 5

6 be two scores for the same subject (first as a control and then as a treatment) or two scores for matched pairs of subjects (one in the control group and one in the treatment group). c) Assumptions: Samples are randomly selected. Samples are representative of the populations for the variable of interest. The distribution of the difference scores in the populations is symmetric around the median of the population for difference scores. 2. Binomial Sign Test for Two Dependent Samples a) What it tests: Do two dependent samples represent two different populations? b) Limitations: Only two samples. You need two scores. This test is based on whether the subject s (or matched pairs of subjects) score increases or decreases by the sign (positive or negative). You can use this test with the assumption of symmetric distribution for the Wilcoxon Matched Paris Test is violated. c) Assumptions: Samples are randomly selected. Samples are representative of the populations for the variable(s) of interest. 3. Friedman Two-Way Analysis of Variance by Ranks a) What it tests: In a group of any number of dependent samples (three, five, ten), do at least two of the samples represent populations with different median values? b) Additional procedures: Like the parametric ANOVA, the Kruskal-Wallace test does not tell you which of the means differed. You must perform pairwise comparisons to determine where the differences lie. See a good statistics book to learn how to do this. You can use the Wilcoxon matched pairs signed ranks test or the binomial sign test for two dependent samples. See a statistics book to learn how to do this. c) Assumptions: Samples are randomly selected. Samples are representative of the populations for the variable(s) of interest. The original variable that was measured was a continuous random variable (this assumption is often violated no idea if that s OK or not, but Sheskin does not seem to think it is a big deal). III. Nominal (Categorical) Data Nonparametric Tests A. NONE TESTS OF DISPERSION I. Interval or Ratio Data - Parametric Tests Variance A. Single sample 1. Single Sample Chi-Square Test for Population Variance a) What it tests: Does a sample come from a population in which the variance equals a known value? b) Limitations: You must know the variance of the population. c) Assumptions: The sample was selected randomly. The sample is representative of the population for the variable of interest. The population is normally distributed with regard to the variable of interest. Statistics 6

7 B. Two or more independent samples 1. Hartley s F(max) Test for Homogeneity of Variance a) What it tests: Are the variances of two or more populations equal? b) Assumptions: The samples were selected randomly. The samples are representative of the populations for the variable of interest. The population is normally distributed with regard to the variable of interest. Sample sizes should be equal or approximately equal. C. Two or more dependent samples 1. The t test for Homogeneity of Variance for Two Dependent Samples a) What it tests: Are the variances of two populations equal? b) Assumptions:The samples were selected randomly. The samples are representative of the populations for the variable(s) of interest. The populations are normally distributed for the variable(s) of interest. II. Ordinal or Rank Ordered Data - Nonparametric Tests Variability A. Single samples NONE B. Two or more independent samples 1. The Siegel-Tukey Test for Equal Variability a) What it tests: Do two independent samples represent two populations with different variances? b) Limitations: You must know or be willing to make some assumptions about the medians of the two populations (see assumption 3 below). c) Assumptions: The samples were randomly selected. They are representative of the populations for the variable(s) of interest and they are independent. The samples represent populations with equal medians for the variable of interest. If you know the medians of the populations and they are not equal, you can perform some adjustments and still use this test. If you do not know the medians and you are unwilling to assume they are equal (probably normally the case), do not use this test. 2. Moses Test for Equal Variability a) What it tests: Do two independent samples represent two populations with different variances? b) Limitations: The data for the dependent variable must have been interval or ratio data originally that were later transformed to ordinal data and the dependent variable must have been a continuous variable (not discrete). c) Assumptions: The samples were randomly selected. The samples are independent and representative of the populations. The original data for the dependent variable were interval or ratio data (they were transformed to ordinal data later). The original data for the dependent variable were continuous (could assume any value). The distribution of two or more populations must have the same general shape (although it need not be normal). Statistics 7

8 C. Two or more dependent samples 1. None that I know III. Nominal (Categorical) Data A. None no adequate measures of variability I. Interval or Ratio Data Parametric Tests A. One Sample Tests of Distribution 1. Single Sample Test for Evaluating Population Skewness a) What it tests: Does the sample come from a population distribution that is symmetrical (not skewed) for the variable of interest? b) Limitations: None c) Assumptions: The sample is representative of the population for the variable of interest. The sample was randomly selected. 2. Single Sample Test for Evaluating Population Kurtosis a) What it tests: Does the sample come from a population distribution that is mesokurtic (not peaked) for the variable of interest? b) Limitations: None c) Assumptions: The sample is representative of the population for the variable of interest. The sample was randomly selected. 3. D Agostino-Pearson Test of Normality a) What it tests: Does the sample come from a population that is normally distributed for the variable of interest? b) Limitations: None c) Assumptions: The sample is representative of the population for the variable of interest. The sample was randomly selected. B. Two or More Independent Samples 1. Use the Single Sample Test for Evaluating Population Skewness, the Single Sample Test for Evaluating Population Kurtosis and the D Agostino-Pearson Test of Normality for each sample. C. Two or More Dependent Samples 1. Use the Single Sample Test for Evaluating Population Skewness, the Single Sample Test for Evaluating Population Kurtosis and the D Agostino-Pearsone Test of Normality for each sample. Statistics 8

9 II. Ordinal (Rank Order) Data Nonparametric Tests A. One Sample 1. Kolmogorov-Smirnov Goodness-of-Fit Test for a Single Sample a) What it tests: Does the distribution of scores in a sample conform to a specific theoretical or empirical (known) population distribution for the variable of interest? b) Limitations: You must know the distribution of the population for the variable of interest. This can be a theoretical distribution (such as the normal distribution) or an empirical (real) distribution. The dependent variable must be continuous (not discrete). This test takes a continuous variable and converts the data into a cumulative frequency (hence it becomes nonparametric data) but you must start with a continuous variable. c) Assumptions: The samples were randomly selected. The samples are independent and representative of the populations for the variable of interest. The original data for the dependent variable were continuous (could assume any value). 2. Lillefor s Test for Normality a) What it tests: Does the distribution of scores in a sample conform to a population distribution for which either the mean or the standard deviation (or both) must be estimated for the variable(s) of interest (an unknown distribution)? b) Limitations: The dependent variable must be continuous (not discrete). This tests takes a continuous variable and converts the data into a cumulative frequency (hence it becomes nonparametric data) but you must start with a continuous variable. c) Assumptions: The samples were randomly selected. The samples are independent and representative of the populations for the variable of interest. The original data for the dependent variable were continuous (could assume any value). B. Two or More Independent Samples 1. Use the Kolmogorov-Smirnov Goodness-of-Fit Test or the Lillefor s Test for Normality for each sample if the data for the dependent variable are continuous. 2. Use the Chi-Square Goodness-of-Fit or the Binomial Sign Test for each sample if the data for the dependent data are NOT continuous. C. Two or More Dependent Samples 1. Use the Kolmogorov-Smirnov Goodness-of-Fit Test or the Lillefor s Test for Normality for each sample if the data for the dependent variable are continuous. 2. Use the Chi-Square Goodness-of-Fit or the Binomial Sign Test for each sample if the data for the dependent data are NOT continuous. Statistics 9

10 III. Nominal (Categorical) Data A. Single Sample 1. Chi-Square Goodness of Fit Test a) What it tests: Are the observed frequencies different from the expected frequencies? b) Limitations: You must know the expected frequency for each category of Responses for the variable of interest. This can either be based on a theoretical (probability-based) distribution or based on some pre-existing empirical information about the variable you are measuring. c) Assumptions: The sample was randomly selected and represents the population for the variable of interest The categories are mutually exclusive. Each observation is represented only once in the data set. The expected frequency of each cell is five or greater. 2. Binomial Sign Test for a Single Sample a) What it tests: Are the observed frequencies for two categories different from the expected frequencies for the variable of interest? b) Limitations: You must know the expected frequency for each category of responses. This can either be based on a theoretical (probability-based) distribution or based on some pre-existing empirical information about the variable you are measuring. This test is just like the Chi-Square Goodness-of-Fit test, but is used for small sample sizes (cell frequency does not have to be five or greater). c) Assumptions: The sample was randomly selected and represents the population for the variable of interest. The categories are mutually exclusive. Each observation is represented only once in the data set. 3. Chi-Square Test of Independence a) What it tests: Is there a relationship between two variables measured for the same sample; e.g., does response for one of the variables predict response for the other variable? b) Limitations: Does not work for very small samples (see assumptions). c) Assumptions: The sample was randomly selected and represents the population for both variables of interest. The categories are mutually exclusive. Each observation is represented only once in the data set. The expected frequency of each cell is five or greater. B. Two or More Independent Samples 1. Chi-Square Test for Homogeneity a) What it tests: Whether or not two or more samples are homogeneous with respect to the proportion of observations in each category of response for the variable of interest. b) Limitations: Does not work for very small samples (see assumptions) c) Assumptions: The sample was randomly selected and represents the population for the variable of interest. The categories are mutually exclusive. Each observation is represented only once in the data set. The expected frequency of each cell is five or greater. Statistics 10

11 2. Fisher Exact Test a) What it tests: Whether or not two or more samples are homogeneous with respect to the proportion of observations in each category of response for the variable of interest. b) Limitations: None commonly used to replace the Chi-Square Test for Homogeneity for small samples. c) Assumptions: The sample was randomly selected and represents the population for the variable of interest. The categories are mutually exclusive. Each observation is represented only once in the data set. C. Two or More Dependent Samples 1. McNemar Test a) What it tests: Do two dependent samples represent two different populations? b) Limitations: Only two samples. Not good for small samples. c) Assumptions: The samples were randomly selected and are representative of the population for the variable(s) of interest. Each observation is independent of all other observations. The scores (dependent variable data) are dichotomous. Categories are mutually exclusive. 2. Cochran Q Test a) What it tests: Among several (three, five, whatever) different samples, do at least two of them represent different populations? b) Limitations: This test does not tell you which samples differed. You must perform additional tests to determine where the differences lie. You can use the McNemar Test to make pairwise comparisons. c) Limitations: Not good for small samples. The samples were randomly selected and are representative of the populations for the variable(s) of interest. Each observation is independent of all other observations. The scores (dependent variable data) are dichotomous. Categories are mutually exclusive. I. Interval or Ratio Data Parametric Tests A. Bivariate Measures MEASURES OF ASSOCIATION 1. Pearson Product-Moment Correlation Coefficient a) What it tests: Is there a significant linear relationship between two variables (predictor or independent and outcome or dependent) in a given population? b) Other calculations needed: The size of the Pearson correlation coefficient (r) in by itself may or may not indicate a statistically significant relationship between the predictor variable. and the outcome variable. At a minimum, you should use a Table of Critical Values for Pearson r and report this value when you use this statistic. The values are different for one-tailed and two-tailed hypotheses. Large r values can be meaningless. Alternatively, small values can be meaningful! You may also need to conduct one or more other tests for evaluating the meaning of the coefficients. Failure to take this step is common and makes many presentations of measures of association fairly useless. The r value alone is not enough! Statistics 11

12 c) Limitations: This is a bivariate measure only two variables, one independent and one dependent variable d) Assumptions: The sample was randomly selected and represents the population. The two variables have a bivariate normal distribution each of the two variables and the linear combination of the two variables are normally distributed. The relationship between the predictor and outcome variables is of equal strength across the whole range of both variables (homoscedasticity). There is no autocorrelation between the two variables. B. Multivariate Measures 1. Multiple Correlation Coefficient a) What it tests: Is there a significant linear relationship between two or more predictor variables and an outcome variable in a given population? b) Other calculations needed: The size of the multiple correlation coefficient (R) by itself may or may not indicate a statistically significant relationship between predictor variables and the outcome variable. At a minimum, you should compute the R2 statistic the coefficient of multiple determination. Then compute the F statistic for R. Use a Table of the F Distribution to determine significance. Large R values can be meaningless. Alternatively, small values can be meaningful! You may also need to conduct one or more other tests for evaluating the meaning of the coefficient. Failure to take this step is common and makes many presentations of measures of association fairly useless. The R or R2" value alone is not enough! c) Limitations: Although you can use a large number of predictor variables, the additional predictive power gained from adding more variables to the model decreases greatly after a few good predictors have been identified. d) Assumptions: The sample was randomly selected and represents the population for the variables of interest. The variables have a bivariate normal distribution each of the variables and the linear combination of the variables are normally distributed. The relationship between the predictor and outcome variables is of equal strength across the whole range of both variables (homoscedasticity). There is no multicollinearity between the predictor variables they are not strongly correlated with each other. 2. Partial Correlation Coefficient a) What it tests: What is the strength of the relationship between one of several predictor variables? Put another way, you hold the values for all but one predictor variable constant and then measure the strength of the one variable that interests you. It is sort of the reverse of multiple correlation. b) Other calculations needed: The size of the partial correlation coefficient (r) by itself may or may not indicate a statistically significant relationship between the predictor variable and the criterion variable. At a minimum, you should compute the value for t and then use a Table of Student s t Distribution to determine significance. The values vary for one-tailed and two-tailed hypotheses. Large r values can be meaningless. Alternatively, small values can be meaningful! You may also need to conduct one or more other tests for evaluating the value of the coefficients. Failure to take this step is common and makes many presentations of measures of association fairly useless. The r value alone is not enough! c) Assumptions: The sample was randomly selected and represents the population with regard to the variables of interest. The variables have a bivariate normal Statistics 12

13 distribution each of the variables and the linear combination of the variables are normally distributed. The relationship between the predictor and outcome variables is of equal strength across the whole range of both variables (homoscedasticity). II. Ordinal or Rank Order Data Nonparametric Measures A. Bivariate Measures 1. Spearman s Rank-Order Correlation Coefficient a) What it tests: In a sample from a population is there a correlation (relationship) between subjects scores on two different variables? Put another way, does a test subject s score for Variable 1 (X) predict his/her score for Variable 2 (Y)? b) Other calculations needed: The size of the Spearman s rank-order correlation coefficient (rs) or Spearman s Rho by itself may or may not indicate a statistically significant relationship between the two variables. You use a Table of Critical Values for Spearman s Rho to determine significance. There are equations you can use, too, one of which gives a t value and one of which gives a z value. The values vary for one-tailed and two-tailed hypotheses. Large rs values can be meaningless. Alternatively, small values can be meaningful! You may also need to conduct one or more other tests for evaluating the value of the coefficients. Failure to take this step is common and makes many presentations of measures of association fairly useless. The rs value alone is not enough! c) Limitations: Only two variables d) Assumptions: The sample was randomly selected and represents the population with regard to the variables of interest.the relationship between the predictor and outcome variables is of equal strength across the whole range of both variables (homoscedasticity). B. Multivariate Measures 1. Kendall s Coefficient of Concordance a) What it tests: In a sample from a population is there a correlation (relationship) between subjects scores on three or more different variables? Put another way, does a test subject s score for Variables 1, 2, 3... (X1, X2, X3...) predict his/her score for Variable 2 (Y)? b) Other calculations needed: The size of the Kendall s coefficient of concordance (W) by itself may or may not indicate a statistically significant relationship between the two variables. You use a Table of Critical Values for Kendall s Coefficient of Concordance to determine significance. You can also computer the significance using the Chi-square statistic and a Table of the Chi-Square Distribution. The values vary for one-tailed and two-tailed hypotheses. Large W values can be meaningless. Alternatively, small values can be meaningful! You may also need to conduct one or more other tests for evaluating the value of the coefficients. Failure to take this step is common and makes many presentations of measures of association fairly useless. The W value alone is not enough! c) Assumptions: The sample was randomly selected and represents the population. The relationship between the predictor and outcome variables is of equal strength across the whole range of both variables (homoscedasticity). Statistics 13

14 III. Nominal or Categorical Data Nonparametric A. There are several measures of association for nominal or categorical data. They are all related to the Chi-Square Test for Homogeneity. They include the Contingency Coefficient, the PhCoefficient, Cramer s PhCoefficient, Yule s Q and the Odds Ratio. All of these measure the degree to which frequencies in one cell of a contingency table are associated with frequencies in other cells or categories that is, the degree of association between the two variables. These measures provide you with precise information about the magnitude of the treatment effect. The Contingency Coefficient can be applied to more than two variables. The PhCoefficient, Cramer s PhCoefficient and Yule s Q can only be used for two variables. The Odds Ratio can be used for more than two variables, but usually is not because it becomes difficult to interpret the results. See a good statistics book if you need to use these measures. Statistics 14

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