Intermediate Statistics Exam 3 (Chapters 15-20/Everything)

Leverage

- Helps to identify outliers based on the predictor variables X(1), X(2), and so on. - Possible values for leverage range from 1/N to 1.0, with a mean of [ (P+1)/N ] - Some recommend carefully inspecting observations with leverage values of [ 3(P+1)/N ] or greater. - Leverage = (Predicted X) - (Mean Predictor Variable)

Box's M criterion

- If a test of Box's M is not significant, the interactions for pairs of RM levels can be calculated by averaging across the different independent groups before testing for sphericity with Mauchly's W statistic. - If Box's M statistic is significant, the mixed design ANOVA methods are not justified, it may be safer to perform separate one-way RM ANOVAs for each of the independent groups.

Beta (β) Coefficients

- Standardized coefficients for the predictor variables. - Account for the fact that there is shared correlation between the IVs and assigns "weights" to each of them.

Articulate the assumptions for using the chi-square test.

- The categories are mutually exclusive and exhaustive. (No observation can be in more than one category; Every possible category must be measured - sometimes researchers fail to consider "non-occurrences" or "no" responses as a separate category. - The observations must be independent (i.e., only one response per subject) - the total number of observations should equal the total number of subjects. - Normality (small expected frequencies (< 5) often create a problem for normality)

Goodness-of-fit Hypothesis Test

- The chi-square test that compares the outcome of a formula to what would be expected by some null hypothesis. - With one factor and many categories in that factor, this test is called a "goodness-of-fit" test (i.e., do the observed frequencies "fit" the expectation. - the degrees of freedom for this test is the number of categories minus one: (df = k - 1)

Chi-square Distribution

- The chi-square test was developed to ascertain the discrepancy between observed frequencies of multinomial categories and the expected frequencies from those categories. - The distribution varies as a function of the df, and actually changes shape noticeably as the df increases. - The distribution is skewed positively, except for very large df. - Large discrepancies between expected and observed frequencies produce large chi-squared statistics and will fall in the positive right tail of the distribution. - The chi-square distribution is an approximation to the multinomial distribution (multinomial = more than two variables)

Understand the differences between the between-subjects and repeated measures version of the ANOVA.

- The conceptual difference is that repeated measures designs allow for the separate estimation of the influence of individual differences from participant to participant, whereas between-subject designs do not. - The repeated measures design is also more economical and contains more statistical power as compared to the between-subject design.

Partial Correlation

- The correlation between the residual of one relationship and the residual of another in the variance of Y. - The correlation of predictor 1 with Y when predictor 2's overlap with Y and X1 is CONTROLLED. - Will usually be larger than the semi-partial correlation (it can be equal but it will never be smaller) - When partialling out variables, you lose one degree of freedom in your significance test for each variable partialled out - this is less of a problem for large samples, but is a consideration when sample size minus the number of variables starts to fall below 30.

Regression Coefficient (b)

- Unstandardized beta coefficients - Basically our slope - b = B/(Sy/S)

Forward Solution

- Can be used so that previously added predictors are not removed at any future steps.

Observed Frequencies

- Data you actually collected. What actually happened in the study. Basically the sums of the nominal data (e.g., 11 yeses and 5 no's).

Regression Diagnostics

- Involve identifying potential outlying observations to help refine the prediction equation.

Be able to compute various measures of association for the chi-square test.

- Phi and Cramer's Phi

Contingency Coefficient

- Related to Cramer's Phi, but people don't use it as much because it doesn't generally range from 0 - 1.

Regression Constant (α)

- The Y intercept - it is the value of the criterion when are the predictors are at zero - In the standardized equation, the regression constant drops out because it is always zero.

Standard Error of b(i)

- The bottom of the t test statistic - Do not need to know how to calculate it - In the output of the relapse regression example

Chi-square Statistic

- The discrepancy between what is expected and what is observed and whether that discrepancy is significant or not. - The larger the discrepancy, the more likely it is to be significant. - df = k - 1 K = categories

Multiple Linear Regression

- The process of fitting a linear combination of multiple X variables to predict scores on Y

Variance Inflation Factor

- The reciprocal of tolerance (1/Tolerance) - Indicates whether a predictor has a strong linear relationship with the other predictor(s) - Represents how much of the standard error of a given predictor coefficient is made larger because of high correlations with other predictors. - We want low standard errors for coefficients, and therefore low VIF values (values over 10 or so might cause concern). - VIF is a collinearity diagnostic statistic provided by SPSS - If eliminating a predictor variable with high VIF and low tolerance helps to generate a more stable model (i.e., low sampling error), then it may be a good idea to do so

Subject x Repeated Measures (RM) factor interaction

- The subject x treatment interaction - A relatively small subject x treatment (or RM) interaction indicates a consistent pattern from subject to subject. (i.e., different subjects are responding in the same general way to the different treatment conditions)

MSresidual

- The sum of both the true amount of subject x treatment interaction and the error. - It is also referred to MSinter (SEE ABOVE) - It is the denominator of the F-ratio

Tests of Independence (or association)

- The two-way chi-squared test. - With two variables, one sets up a contingency table to see if the frequencies of one factor are contingent (i.e., depend) on the other

Expected Frequencies

- Tied to the null hypothesis (e.g., what you expect to happen - 8 yeses and 8 no's)

Variability (SS) due to subjects

- We can't control this because it is due to individual differences. - This is taken into account, but then ignored in the computation of F - Also called SS(row) on formula sheet

Least Squares Criterion

- When our "weights" of our predictor variables give us the highest possible R, we have minimized the sum of the squared errors from our predictions.

MSinter

- calculated by = SS(subxRM)/df(subxRM) - df(subxRM) = (K - 1)(n - 1) - K = groups - n = subjects in each group

Homogeneity of covariance across groups (mixed design)

- An assumption unique to the mixed designs - It states that the covariance structure among the RM levels must be the same for all of the independent groups. - This can be tested in terms of a statistic known as Box's M criterion.

Effect size for RM designs

- An effect size measure, "f" , is used to represent the population SD between groups from the grand mean, versus population SD within a group. - Labeled "Est. f" on formula sheet - .10 is "small" - .25 is "medium" - .40 is "large"

Cramer's Phi

- A measure of association for two-way tables other than 2 x 2 - Similar to any other correlation, squaring it is analogous to omega-squared and r-squared - "L" in the formula refers to the number of rows or columns, whichever is smaller - Listed as "Cramer's V" in SPSS

Distance

- Also called discrepancy - Refers to the error in prediction in the criterion variable. - This error could be random error, could be incorrectly recorded, or could be an unusual case that is justified in being thrown out. - Distance = (Actual score on Y) - (Predicted Y)

Understand how the concept of multiple linear regression is similar, and is different from, bivariate linear regression.

- Bivariate linear regression only compares one variable with one predictor, whereas multiple linear regression compares one variable with more than one predictor variable. - ADD more - how are they similar?

Influence

- Combines distance and leverage for a given observation to ascertain whether that observation markedly changes the regression surface (i.e., the equation). - Most common measure is Cook's D - Cook's D reflects the change in a regression coefficient if the offending observation were taken out and the regression re-computed. - Values over 1.0 are highly unusual and merit scrutiny. - A more conservative rule of thumb is: Cook's D > [ 4/(N - k - 1) ] k = number of predictors (can also be "P")

Compound symmetry and sphericity

- Compound Symmetry - must have the equal variances and covariances. This is a highly restrictive assumption and most agree that sphericity is enough. - When compound symmetry exists, the population correlation (p) between any pair of treatment levels is the same as between any other pair. - Sphericity - the variances of difference scores from all possible pairs of conditions are equal. Want the Test of Sphericity in SPSS to NOT be significant. - Low sphericity tends to increase Type I Errors (i.e., false positive).

Multiple Correlation Coefficient (R)

- The correlation between your predictors for some criterion (based on two or more predictors) and the actual values for that criterion. - Multiple R is a correlation coefficient that represents the entire linear combination of predictors, rather than the simple correlation of given X variable with Y. - Calculated by adding the squared r's and then take the square-root

Two-way Contingency Table

- With two variables in the chi-square test, one sets up a contingency table to see if the frequencies of one factor are contingent (i.e., depend) on the other.

Explain how we use regression diagnostics to identify potential data points that have an undue influence on the regression equation.

- Consists of distance, leverage, and influence. - Regression diagnostics involve identifying potential outlying observations to help refine the prediction equation.

Part (semi-partial) Correlation

- Correlation of one predictor variable with Y, that is independent of the other predictor variable. - This correlation tells us how strongly related predictor 1 is with Y, when predictor 2 is not allowed to vary. - The squared semi-partial correlation is one way to determine how much influence (i.e., importance) each predictor has to the full equation.

Categorical (Nominal) Scale

- Data is always in nominal scale for chi-square - Data is neither numbered, nor ordered, but subjects are allocated to distinct categories

Homogeneity of covariance

- Does not apply when the groups are independent or when the design only has two treatment levels.

Epsilon (degree of sphericity) (ε)

- Epsilon is an estimate of the degree of sphericity in the population. - It ranges from a maximum of 1.0 when sphericity holds, down to a minimum of 1/(c - 1) when the data suggests a total lack of sphericity. - The ordinary df components from the RM ANOVA are multiplied by ε, and then the adjusted df components can be used to find the adjusted critical value of F.

Multicollinearity

- Exists when there is a strong correlation between two or more predictors in the model. - As collinearity increases, so do the standard errors of the b coefficients - Multicollinearity makes it difficult to assess the individual importance of a predictor

Understand how calculating expected frequencies may differ depending on the type of test (goodness-of-fit vs. association) and the null hypothesis itself (expected frequencies are equal vs. in some proportion to one another).

- Expected frequencies (E) for goodness of fit is based on the null hypothesis, so assuming no relationship, each choice would be equal to one another. - For association, E is found by taking the total number for each column and total number for each row, multiplying them, then dividing that number by the total number of subjects in the study.

Understand how the chi-square distribution is used as the underlying distribution for hypothesis tests on categorical data.

- For the hypothesis test for chi-square, we don't use the normal distribution, we use the chi-square distribution. - The chi-square distribution looks positively skewed unless the df is very large, and even then it is only because the mean is large and there are no negative values.

Be able to compute a goodness-of-fit test and the test of association.

- Goodness-of-fit is a one-way - With the goodness-of-fit test, you are measuring the "fit" between observed frequencies and expected frequencies (i.e., frequencies predicted by a theory or a hypothesized population distribution). - Test of association is a two-way - The only difference in computation is how you find the expected value (E) and the df.

Understand how we can tackle post hoc analyses based on whether sphericity is tenable or not (e.g., the LSD when sphericity is violated).

- If sphericity is assumed - we can proceed with the multiple comparisons by any of the methods described in Chapter 13, substituting MS(inter) for MS(w) (e.g., the LSD) - If we can't assume sphericity - the use of the error term from the overall analysis (MS(inter)) is not justified for each post hoc comparison. - Instead: - The simplest/safest procedure for pairwise comparisons is to perform the ordinary matched t tests for each pair of conditions using a Bonferroni adjustment to control experiment-wise alpha.

Backward Solution

- Involves putting all predictors into the model and then removing those that contribute the least to the model. - This is done iteratively until we are left with the model that only includes influential predictors.

Hyperplane

- Just as regression with one predictor involves finding a straight line that minimizes squared errors of the predictions, regression with two predictors amounts to finding the best fitting (two-dimensional) regression plane. - This hyperplane has two partial regression slopes, and the degrees of freedom are reduced by the number of partial slopes, plus one for the intercept (df = N - P - 1).

Measures of Agreement

- Kappa is a type of measuring agreement - Kind of like a version of IOA

Phi Coefficient

- Measure of association for 2 x 2 tables specifically. - Similar to any other correlation, squaring it is analogous to omega-squared and r-sqaured

Omega-squared for RM designs

- Measure of the association between the IV and the DV - .01 is "small" - .06 is "medium" - .15 is "large" - If there is a high correlation among the measures, omega-squared will also be high

Coefficient of Determination

- Multiple R-squared - The amount of variance in Y shared by the linear combination of predictors. - Can overestimate the true amount of shared variance when there are numerous predictors and a small number of observations - an adjustment for this problem is Adjusted R-Squared

Explain what the adjusted R-squared statistic adjusts and in what situations it will deviate substantially from the unadjusted R-squared.

- Multiple R-squared can overestimate the true amount of shared variance when there are numerous predictors and a small number of observations, so Adjusted R-squared accounts for this. - If the sample size is big, minimal adjustments will take place for Adjusted R-squared, but with a small sample size, there will be a substantial adjustment.

Adjusted R-squared

- Multiple R-squared can overestimate the true amount of shared variance when there are numerous predictors and a small number of observations. - The Adjusted R-squared accounts for this problem.

Linear Combination

- Multiple R-squared is the amount of variance in Y shared by the linear combination of predictors.

Be able to calculate effect size measures and consider power and estimated sample size to plan for power.

- Once you get "est. f" for effect size, then you can use the non-centrality parameter for the F distribution, labeled phi, to estimate power. - Φ = f x (square root of n) - n = (Φ / f)^2 - Can consult table A.10 to obtain the values for phi, based on the number of conditions (k) and the likely df(res)

Understand the difference between part (semi-partial) and partial correlation, and how they relate to the use of regression.

- Part (semi-partial) - correlation of Y with a specific predictor that is independent of the other predictor. - Partial - correlation of the residual of one relationship and the residual of the of another in the variance of Y. - These are a way to determine the influence of each variable.

Understand the properties of the multiple correlation coefficient (R) and how it differs from the standard bivariate correlation coefficient (r).

- R represents the entire linear combination of predictors, rather than a simple correlation of given X variable with Y. - R ranges from 0 to 1 - R also represents the correlation between each Y score and its associated predicted Y score.

Be able to test whether multiple R is different for different versions of a regression model when one model is a subset of a larger model.

- Refers to hierarchical regression, basically saying that when you do one of those, there's a new model, therefore a new R. - In the model summary output, each row is a new model so you have to look at how much it changes between each one.

Tolerance

- Refers to the degree to which one predictor can itself be predicted by the remaining predictors in the model. The predictor is compared against all other predictors in the model sequentially - It tells us the degree of overlap among predictors, lower values of tolerance represent potential problems (lower than .2 or so is a flag). - We want multiple correlation among predictors to be rather low, and therefore tolerance to be rather high.

Hierarchical Regression

- Regression procedure when the ordering of variables is based on a theoretical model. - This form of regression uses many of the tools developed for prediction but involves a greater exercise of judgment guided by theory and knowledge of the area of research.

Understand how SSsub is calculated and ignored as a source of error variability.

- SS(sub) is calculated by taking every subjects mean and subtracting the grand mean. Then that number is squared. This is done for every subject and then summed together. It is then multiplied by the number of groups (i.e., treatment conditions). - It is then ignored in through the calculation of SS(subxRM) = SS(T) - SS(RM) - SS(sub) - SSsub is ignored because it is individual differences.

Explain how different procedures affect how predictors are placed into a multiple regression equation (e.g., backward, stepwise), when these procedures are most useful, and which is most recommended.

- Stepwise is the most recommended, but it also depends on the data. - They are useful at being able to partition the degree of influence of all the predictors.

Stepwise Solution

- The opposite of backward elimination, where we add the most influential predictor, then the next most, and so on. - At each step in the process, before adding a new variable, we determine whether any current predictors should be removed on the basis that they no longer make a contribution (e.g., p < .05). - Selection stops when remaining predictors will not make a contribution.

Understand how we can deal with violations of sphericity.

- There are different ways that involve applying a correction for the Epsilon value. - Lower bound correction - this is a change to the critical F value from df = K-1, (n - 1)(K-1) to df = 1, n - 1 (This increases the critical F value and is extremely conservative) - The Huynh & Feldt and Geisser-Greenhouse are corrections to the df based on the degree of violation to sphericity, and create more modest corrections to the critical F value - The non-corrected df are multiplied by the epsilon values for each respective procedure.

Be able to recognize and use a multiple prediction equation, both in unstandardized and standardized form.

- Yhat equation (undstandardized) - Z(Yhat) equation (standardized)

Understand how to calculate eta-squared RM (which is a partial measure) and how that differs from eta-squared based on SStotal.

- eta-squared RM ( or n^2 RM) = SS(RM)/SS(RM) + SS(inter) - It tends to overestimate the impact that an independent variable would have when tested with independent groups, or considerably less effective matching. - It is also rather specific to the experimental design of the given study. - Ordinary eta-squared is calculated as though the groups were independent (n^2 = SS(RM)/SS(Total) - This equals SS(RM)/(SS(RM) + SS(sub) + SS(inter) - Usually eta-squared will not be as large as eta-squared RM, but it will provide a more stable basis for planning future experiments.

Be able to compute the repeated measures ANOVA, using the steps of hypothesis testing.

1) State the Hypothesis. - It is the same as for one-way independent groups ANOVA: - Null: H(0): u(1) = u(2) = u(3) - Alternative: H(A): H(0) is not true 2) Select the Statistical Test and Significance Level. - RM ANOVA 3) Select the Samples and Collect the Data. 4) Find the Region of Rejection. - df for the numerator is the same as in the independent ANOVA (one less than the number of treatments) - df for the denominator is the df for the subject x treatment interaction (one less than the number of subject times one less the number of treatments) 5) Calculate the Test Statistic. 6) Make the Statistical Decision.