CHOOSING AN APPROPRIATE BIVARIATE INFERENTIAL …



Choosing an Appropriate Inferential StatisticFor each variable, you must decide whether it is, for practical purposes, categorical (only a few values are possible) or continuous (many values are possible). K = the number of values of the variable.Parametric, Nonparametric, and Resampling ProceduresThe parametric procedures are usually a bit more powerful than the nonparametrics if their assumptions are met. Thus, you should use a parametric test if you can. The nonparametric test may well have more power than the parametric test if the parametric test’s assumptions are violated, so, if you cannot meet the assumptions of the parametric test, using the nonparametric should both keep alpha where you set it and lower beta. Resampling statistics (such as bootstrapping) are becoming a popular alternative to parametric statistics when the assumptions of the latter cannot be met.Only One VariableYou wish to predict the value of continuous Y for a randomly selected member of the population from which your data were sampled. You wish to do so in such a way that the error in prediction is as small as possible, using the least squares criterion -- is as small as possible. This is actually the most simple linear regression model – the model has only one parameter, -- that is, . For each case, the predicted score will be the sample mean.If your desire were to use a predictor that minimized the sum of the absolute deviations between actual Y and predicted Y, , your model would be predicted Y = the sample median of Y.If you wish to predict the value of categorical variable Y, your predictor should be the mode.Both Variables CategoricalThe Pearson chi-square is appropriately used to test the null hypothesis that two categorical (K 2) variables are independent of one another.If each variable is dichotomous (K = 2), the phi coefficient ( ) is also appropriate. If you can assume that each of the dichotomous variables measures a normally distributed underlying construct, the tetrachoric correlation coefficient is appropriate.Two Continuous VariablesThe Pearson product moment correlation coefficient (r) is used to measure the strength of the linear association between two continuous variables. To do inferential statistics on r you need assume normality (and homoscedasticity) in (across) X, Y, (X|Y), and (Y|X).Linear regression analysis has less restrictive assumptions (no assumptions on X, the fixed variable) for doing inferential statistics, such as testing the hypothesis that the slope of the regression line for predicting Y from X is zero in the population.The Spearman rho is used to measure the strength of the monotonic association between two continuous variables. It is no more than a Pearson r computed on ranks and its significance can be tested just like r.Kendall’s tau coefficient (), which is based on the number of inversions (across X) in the rankings of Y, can also be used with rank data, and its significance can be tested.Categorical X, Continuous YYou need to decide whether your design is independent samples (no correlation expected between Y at any one level of X and Y at any other level of X, also called between subjects or Completely Randomized Design—subjects randomly sampled from the population and randomly assigned to treatments) or correlated samples. Correlated samples designs include the following: Within-subjects (also known as “repeated measures”) matched pairs, and randomized blocks. In within-subjects designs each subject is tested (measured on Y) at each level of X. That is, the (third, hidden) variable Subjects is crossed with rather than nested within X.I am assuming that you are interested in determining the “effect” of X upon the location (central tendency - mean, median) of Y rather than dispersion (variability) in Y or shape of distribution of Y. If it is variance in Y that interests you, use an FMAX Test (see Wuensch for special tables if K > 2 or for more powerful procedures), Levene’s test, or Obrien’s test, all for independent samples. For correlated samples, see Howell's discussion of the use of t derived by Pitman (Biometrika, 1939). If you wish to determine whether X has any effect on Y (location, dispersion, or shape), use one of the nonparametrics.Independent SamplesFor K 2, the independent samples parametric one-way analysis of variance is the appropriate statistic if you can meet its assumptions, which are normality in Y at each level of X and constant variance in Y across levels of X (homogeneity of variance). You may need to transform or trim or Windsorize Y to meet the assumptions. If you can meet the normality assumption but not the homogeneity of variance assumption, you should adjust the degrees of freedom according to Box or adjust df and F according to Welch.For K 2, the Kruskal-Wallis nonparametric one-way analysis of variance is appropriate, especially if you have not been able to meet the normality assumption of the parametric ANOVA. To test the null hypothesis that X is not associated with location of Y you must be able to assume that the dispersion in Y is constant across levels of X and that the shape of the distribution of Y is constant across levels of X.For K = 2, the parametric ANOVA simplifies to the pooled variances independent samples t test. The assumptions are the same as for the parametric ANOVA. The computed t will be equal to the square root of the F that would be obtained were you to do the ANOVA and the p will be the same as that from the ANOVA. A point-biserial correlation coefficient is also appropriate here. In fact, if you test the null hypothesis that the point-biserial = 0 in the population, you obtain the exact same t and p you obtain by doing the pooled variances independent samples t-test. If you can assume that dichotomous X represents a normally distributed underlying construct, the biserial correlation is appropriate. If you cannot assume homogeneity of variance, use a separate variances independent samples t test, with the critical t from the Behrens-Fisher distribution (use the Cochran & Cox approximation) or with df adjusted (the Welch-Satterthwaite solution).For K = 2, with nonnormal data, the Kruskal-Wallis could be done, but more often the rank nonparametric statistic employed will be the nonparametric Wilcoxon rank sum test (which is essentially identical to, a linear transformation of, the Mann-Whitney U statistic). Its assumptions are identical to those of the Kruskal-Wallis.Correlated SamplesFor K 2, the correlated samples parametric one-way analysis of variance is appropriate if you can meet its assumptions. In addition to the assumptions of the independent samples ANOVA, you must assume sphericity, which is essentially homogeneity of covariance—that is, the correlation between Y at Xi and Y at Xj must be the same for all combinations of i and j. This analysis is really a Factorial ANOVA with subjects being a second X, an X which is crossed with (rather than nested within) the other X, and which is random-effects rather than fixed-effects. If subjects and all other X’s were fixed-effects, you would have parameters instead of statistics, and no inferential procedures would be necessary. There is a multivariate approach to the analysis of data from correlated samples designs, and that approach makes no sphericity assumption. There are also ways to correct (alter the df) the univarariate analysis for violation of the sphericity assumption.For K 2 with nonnormal data, the rank nonparametric statistic is the Friedman ANOVA. Conducting this test is equivalent to testing the null hypothesis that the value of Kendall’s coefficient of concordance is zero in the population. The assumptions are the same as for the Kruskal-Wallis.For K = 2, the parametric ANOVA could be done with normal data but the Correlated samples t-test is easier. We assume that the difference-scores are normally distributed. Again, .For K = 2, a Friedman ANOVA could be done with nonnormal data, but more often the nonparametric Wilcoxon’s signed-ranks test is employed. The assumptions are the same as for the Kruskal-Wallis. Additionally, for the test to make any sense, the difference-scores must be rankable (ordinal), a conditional that is met if the data are interval. A binomial sign test could be applied, but it lacks the power of the Wilcoxon.Categorical X with K > 2, Continuous YIf your omnibus (overall) analysis is significant (and maybe even if it is not) you will want to make more specific comparisons between pairs of means (or medians). For nonparametric analysis, use one of the Wilcoxon tests. You may want to use the Bonferroni inequality (or Sidak’s inequality) to adjust the per comparison alpha downwards so that familywise alpha does not exceed some reasonable (or, unreasonable, like .05) value. For parametric analysis there are a variety of fairly well known procedures such as Tukey’s tests, REGWQ, Newman-Keuls, Dunn-Bonferroni, Dunn-Sidak, Dunnett, etc. Fisher’s LSD protected test may be employed when K = 3.More Than One X and/or More Than One YNow this is multivariate statistics. I include in this category the following:Multidimensional contingency table analysis. This is like the Pearson 2, but you have more than two categorical variables and the tests are based on likelihood ratios. You can test all of the relationship between the categorical variables in the model. One special case of this analysis is the logit analysis, where you are only interested in effects that involve the variable designed as the dependent variable.Multiple regression. More than one predictor (X) variable, but only one Y variable. Bivariate linear regression is a special case of multiple regression and multiple regression is a special case of canonical regression.Least squares ANOVA and ANCOV. Here dummy variables are used to code group membership and the analysis is really a multiple regression analysis. This can be especially useful in factorial designs where the factors are correlated with each other. These analyses are special cases of canonical regression.Binary logistic regression. This employs the Generalized Linear Model, rather than the General Linear Model. The natural log is used to link the Y variable (the odds of being in one of the two groups) to the weighted linear combination of the predictor variables. The model can then be used to predict group membership. This analysis can be enhanced to multinomial logistic regression, with which one can predict membership in three or more groups.Discriminant function analysis. As with logistic regression, your Y variable is group membership. Your predictor variables are typically continuous and nicely distributed. This is a special case of canonical regression.MANOVA and MANCOVA. Multiple analysis of variance/covariance. For the one-way MANOVA, this is identical to a discriminant function analysis, except that group membership becomes the X variable and scores on normally distributed variables become the Y variables. If there two or more grouping variables, the analysis may be factorial, and if the model includes a normally distributed X variable the analysis is a MANCOVA. These analyses are special cases of canonical regression.Canonical correlation/regression. Here the goal is to produce one or more weighted linear combinations of the X variables and one or more weighted linear combinations of the Y variables such that the correlation(s) between those linear combinations (canonical variates) is as large as possible.Hierarchical linear modeling. Aka multilevel linear modeling. This analysis is essentially multiple linear regression but where the data are nested. There will be two or more levels of variables. A typical example involves students nested within schools and those schools nested within school districts. The nesting here is very similar to that in independent samples ANOVA, where subjects is a random effects variable that is nested within (fixed effects) groups. The outcome variables can be modeled in terms of their relationship with variables at all of the levels.Principle components and factor analysis. Here you have only one set of variables, not a set of X variables and a set of Y variables. The usual goal is to reduce a large number of variables to a smaller number of components or factors which capture most of the information (covariance) in the original variables.Structural equation modeling. This is a combination of factor analysis and multiple regression. Weighted linear combinations of the observed variables are created such that they maximize the modeled relationships among the latent variables which are theorized the underlie the observed variables.Copyright 2016, Karl L. Wuensch - All rights reserved. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download