Reporting hierarchical multiple regression apa table

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Reporting hierarchical multiple regression apa table

Arndt Regorz, Dipl. Kfm. & M.Sc. Psychologie, 01/18/2020 If the option "Collinearity Diagnostics" is selected in the context of multiple regression, two additional pieces of information are obtained in the SPSS output. First, in the "Coefficients" table on the far right a "Collinearity Statistics" area appears with the two columns "Tolerance" and "VIF". And below this table appears another table with the title "Collinearity Diagnostics": The interpretation of this SPSS table is often unknown and it is somewhat difficult to find clear information about it. The following tutorial shows you how to use the "Collinearity Diagnostics" table to further analyze multicollinearity in your multiple regressions. The tutorial is based on SPSS version 25. Content 1. YouTube Video-Tutorial (Note: When you click on this video you are using a service offered by YouTube.) 2. Column "Dimension" Let us start with the first column of the table. Similar but not identical to a factor analysis or PCA (principle component analysis), an attempt is made to determine dimensions with independent information. More precisely, a singular value decomposition (Wikipedia, n.d.) of the X matrix is apparently performed without its prior centering (Snee, 1983). 3. Column "Eigenvalue" Several eigenvalues close to 0 are an indication for multicollinearity (IBM, n.d.). Since "close to" is somewhat imprecise it is better to use the next column with the Condition Index for the diagnosis. 4. Column "Condition Index" These are calculated from the eigenvalues. The condition index for a dimension is derived from the square root of the ratio of the largest eigenvalue (dimension 1) to the eigenvalue of the dimension. In the table above, for example, for dimension 3: Eigenvalue dim 1: 6.257 Eigenvalue dim 3: 0.232 Eigenvalue dim 1 / Eigenvalue dim 3: 26.970 Square root (=Condition Index): 5.193 (the difference to the output 5.196 is due to rounding error) More important than the calculation is the interpretation of the Condition Index. Values above 15 can indicate multicollinearity problems, values above 30 are a very strong sign for problems with multicollinearity (IBM, n.d.). For all lines in which correspondingly high values occur for the Condition Index, one should then consider the next section with the "Variance Proportions". 5. Section "Variance Proportions" Next, consider the regression coefficient variance-decomposition matrix. Here for each regression coefficient its variance is distributed to the different eigenvalues (Hair, Black, Babin, &Anderson, 2013). If you look at the numbers in the table, you can see that the variance proportions add up to one column by column. According to Hair et al. (2013) for each row with a high Condition Index, you search for values above .90 in the Variance Proportions. If you find two or more values above .90 in one line you can assume that there is a collinearity problem between those predictors. If only one predictor in a line has a value above .90, this is not a sign for multicollinearity. However, in my experience this rule does not always lead to the identification of the collinear predictors. It is quite possible to find multiple variables with high VIF values without finding lines with pairs (or larger groups) of predictors with values above .90. In this case I would also search for pairs in a line with variance proportion values above .80 or .70, for example. 6. Hierarchical regression If you perform a hierarchical regression, the corresponding values of the "collinearity diagnostics" table appear separately for each regression step ("Model 1", "Model 2"): I would primarily interpret the data for the last step or, in general, the data for those steps that you report and interpret for your hypothesis tests in more detail. 7. How to use the information When I want to analyze a multiple regression output for multicollinearity, this is how I proceed: I look at the value "VIF" in the table "Coefficients". If this value is less than 10 for all predictors the topic is closed for me. If there are only a maximum of two values of the VIF above 10, I assume that the collinearity problem exists between these two values and do not interpret the "collinearity diagnostics" table. However, if there are more than two predictors with a VIF above 10, then I will look at the collinearity diagnostics. I identify the lines with a Condition Index above 15. In these lines I check if there is more than one column (more than one predictor) with values above .90 in the variance proportions. In this case I assume a collinearity problem between the predictors that have these high values. If only one predictor in a line has a high value (above .90), this is not relevant to me. If I have not been able to identify the source of the multicollinearity yet, because there are no lines with several variance proportions above .90, I reduce this criterion and consider, for example, pairs of predictors (or groups of predictors) with values above .80 or .70 as well. 8. Example Step 1: There are predictors with a VIF above 10 (x1, x2, x3, x4). Step 2: There are more than two predictors (here: four) to which this applies. Therefore look at the collinearity diagnostics table: Step 3: Dimensions 6 and 7 show a condition index above 15. Step 4: For each of the two dimensions search for values above .90. For dimension 6 we find these for the predictors x1 and x2, for dimension 7 for the predictors x3 and x4. On this basis you assume that there are actually two different collinearity problems in your model: between x1 and x2 and between x3 and x4. (However, if all values above .90 for these four predictors had been on one line, that would have indicated a single multicollinearity problem of all four variables). Steps 5 and 6 are not used in this example because we have already identified the sources for collinearity. 9. References Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2013). Multivariate data analysis: Advanced diagnostics for multiple regression [Online supplement]. Retrieved from IBM (n.d.). Collinearity diagnostics. Retrieved August 19, 2019, from Snee, R. D. (1983). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. Journal of Quality Technology, 15, 149-153. doi:10.1080/00224065.1983.11978865 Wikipedia (n.d.). Singular value decomposition. Retrieved August 19, 2019, from Additional tutorials about multivariate statistics Posted on Friday, May 20th, 2016 at 8:38 pm.Written by Bommae This post is NOT about Hierarchical Linear Modeling (HLM; multilevel modeling). The hierarchical regression is model comparison of nested regression models. When do I want to perform hierarchical regression analysis? Hierarchical regression is a way to show if variables of your interest explain a statistically significant amount of variance in your Dependent Variable (DV) after accounting for all other variables. This is a framework for model comparison rather than a statistical method. In this framework, you build several regression models by adding variables to a previous model at each step; later models always include smaller models in previous steps. In many cases, our interest is to determine whether newly added variables show a significant improvement in \(R^2\) (the proportion of explained variance in DV by the model). Let's say we're interested in the relationships of social interaction and happiness. In this line of research, the number of friends has been a known predictor in addition to demographic characteristics. However, we'd like to investigate if the number of pets could be an important predictor for happiness. The first model (Model 1) typically includes demographic information such as age, gender, ethnicity, and education. In the next step (Model 2), we could add known important variables in this line of research. Here we would replicate previous research in this subject matter. In the following step (Model 3), we could add the variables that we're interested in. Model 1: Happiness = Intercept + Age + Gender (\(R^2\) = .029) Model 2: Happiness = Intercept + Age + Gender + # of friends (\(R^2\) = .131) Model 3: Happiness = Intercept + Age + Gender + # of friends + # of pets (\(R^2\) = .197, \(\Delta R^2\) = .066) Our interest is whether Model 3 explains the DV better than Model 2. If the difference of \(R^2\) between Model 2 and 3 is statistically significant, we can say the added variables in Model 3 explain the DV above and beyond the variables in Model 2. In this example, we'd like to know if the increased \(R^2\) .066 (.197 ? .131 = .066) is statistically significant. If so, we can say that the number of pets explains an additional 6% of the variance in happiness and it is statistically significant. There are many different ways to examine research questions using hierarchical regression. We can add multiple variables at each step. We can have only two models or more than three models depending on research questions. We can run regressions on multiple different DVs and compare the results for each DV. Conceptual Steps Depending on statistical software, we can run hierarchical regression with one click (SPSS) or do it manually step-by-step (R). Regardless, it's good to understand how this works conceptually. Build sequential (nested) regression models by adding variables at each step. Run ANOVAs (to compute \(R^2\)) and regressions (to obtain coefficients). Compare sum of squares between models from ANOVA results. Compute a difference in sum of squares (\(SS\)) at each step. Find corresponding F-statistics and p-values for the \(SS\) differences. Compute increased \(R^2\)s from the \(SS\) differences. \(R^2 = \frac{SS_{Explained}}{SS_{Total}}\) Examples in R In R, we can find sum of squares and corresponding F-statistics and p-values using anova(). When we use anova() with a single model, it shows analysis of variance for each variable. However, when we use anova() with multiple models, it does model comparisons. Either way, to use anova(), we need to run linear regressions first. # Import data (simulated data for this example) myData

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