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### Strategy for Complete Regression Analysis

Additional issues in regression analysis

Assumption of independence of errors

Influential cases

Multicollinearity

Adjusted R²

Strategy for Solving problems

Sample problems

Complete regression analysis

Assumption of independence of errors - 1

- Multiple regression assumes that the errors are independent and there is no serial correlation. Errors are the residuals or differences between the actual score for a case and the score estimated using the regression equation. No serial correlation implies that the size of the residual for one case has no impact on the size of the residual for the next case.
- The Durbin-Watson statistic is used to test for the presence of serial correlation among the residuals. The value of the Durbin-Watson statistic ranges from 0 to 4. As a general rule of thumb, the residuals are not correlated if the Durbin-Watson statistic is approximately 2, and an acceptable range is 1.50 - 2.50.

Assumption of independence of errors - 2

- Serial correlation is more of a concern in analyses that involve time series.
- If it does occur in relationship analyses, its presence can be usually be understood by changing the sequence of cases and running the analysis again.
- If the problem with serial correlation disappears, it may be ignored.
- If the problem with serial correlation remains, it can usually be handled by using the difference between successive data points as the dependent variable.

Multicollinearity - 1

- Multicollinearity is a problem in regression analysis that occurs when two independent variables are highly correlated, e.g. r = 0.90, or higher.
- The relationship between the independent variables and the dependent variables is distorted by the very strong relationship between the independent variables, leading to the likelihood that our interpretation of relationships will be incorrect.
- In the worst case, if the variables are perfectly correlated, the regression cannot be computed.
- SPSS guards against the failure to compute a regression solution by arbitrarily omitting the collinear variable from the analysis.

Multicollinearity - 2

- Multicollinearity is detected by examining the tolerance for each independent variable. Tolerance is the amount of variability in one independent variable that is no explained by the other independent variables.
- Tolerance values less than 0.10 indicate collinearity.
- If we discover collinearity in the regression output, we should reject the interpretation of the relationships as false until the issue is resolved.
- Multicollinearity can be resolved by combining the highly correlated variables through principal component analysis, or omitting a variable from the analysis.

Adjusted R²

- The coefficient of determination, R² which measures the strength of a relationship, can usually be increased simply by adding more variables. In fact, if the number of variables equals the number of cases, it is often possible to produce a perfect R² of 1.00.
- Adjusted R² is a measure which attempts to reduce the inflation in R² by taking into account the number of independent variables and the number of cases.
- If there is a large discrepancy between R² and Adjusted R², extraneous variables should be removed from the analysis and R² recomputed.

Influential cases

- The degree to which outliers affect the regression solution depends upon where the outlier is located relative to the other cases in the analysis. Outliers whose location have a large effect on the regression solution are called influential cases.
- Whether or not a case is influential is measured by Cook’s distance.
- Cook’s distance is an index measure; it is compared to a critical value based on the formula: 4 / (n – k – 1) where n is the number of cases and k is the number of independent variables.
- If a case has a Cook’s distance greater than the critical value, it should be examined for exclusion.

Overall strategy for solving problems

- Run a baseline regression using the method for including variables implied by the problem statement to find the initial strength of the relationship, baseline R².
- Test for useful transformations to improve normality, linearity, and homoscedasticity.
- Substitute transformed variables and check for outliers and influential cases.
- If R² from regression model using transformed variables and omitting outliers is at least 2% better than baseline R², select it for interpretation; otherwise select baseline model.
- Validate and interpret the selected regression model.

Problem 1

In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

Survey respondents who were older had fewer family members earning money. The variables sex and respondent's socioeconomic index did not have a relationship to how many in family earned money.

1. True

2. True with caution

3. False

4. Inappropriate application of a statistic

Dissecting problem 1 - 1

When we test for influential cases using Cook’s distance, we need to compute a critical value for comparison using the formula:

4 / (n – k – 1)

where n is the number of cases and k is the number of independent variables. The correct value (0.0160) is provided in the problem.

The problem may give us different levels of significance for the analysis.

In this problem, we are told to use 0.05 as alpha for the regression, but 0.01 for testing assumptions.

In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

After evaluating assumptions, outliers, and influential cases, we will decide whether we should use the model with transformations and excluding outliers, or the model with the original form of the variables and all cases.

The random number seed (788035) for the split sample validation is provided.

Dissecting problem 1 - 2

The variables listed first in the problem statement are the independent variables (IVs): "age" [age], "sex" [sex], and

"respondent's socioeconomic index" [sei].

When a problem states that there is a relationship between some independent variables and a dependent variable, we do standard multiple regression.

In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

Survey respondents who were older had fewer family members earning money. The variables sex and respondent's socioeconomic index did not have a relationship to how many in family earned money.

1. True

2. True with caution

3. False

4. Inappropriate application of a statistic

The variable that is the target of the relationship is the dependent variable (DV): "how many in family earned money“ [earnrs].

Dissecting problem 1 - 3In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

In order for a problem to be true, we will have to find that there is a statistically significant relationship between the set of IVs and the DV, and the strength of the relationship stated in the problem must be correct.

Survey respondents who were older had fewer family members earning money. The variables sex and respondent's socioeconomic index did not have a relationship to how many in family earned money.

1. True

2. True with caution

3. False

4. Inappropriate application of a statistic

In addition, the relationship or lack of relationship between the individual IV's and the DV must be identified correctly, and must be characterized correctly.

LEVEL OF MEASUREMENTIn the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

Survey respondents who were older had fewer family members earning money. The variables sex and respondent's socioeconomic index did not have a relationship to how many in family earned money.

Multiple regression requires that the dependent variable be metric and the independent variables be metric or dichotomous.

1. True

2. True with caution

3. False

4. Inappropriate application of a statistic

"How many in family earned money" [earnrs] is an interval level variable, which satisfies the level of measurement requirement.

"Age" [age] and "respondent's socioeconomic index" [sei] are interval level variables, which satisfies the level of measurement requirements for multiple regression analysis.

"Sex" [sex] is a dichotomous or dummy-coded nominal variable which may be included in multiple regression analysis.

The baseline regression

We begin out analysis by runring a standard multiple regression analysis with earnrs as the dependent variable and age, sex, and sei as the independent variables.

Select Enter as the Method for including variables to produce a standard multiple regression.

Click on the Statistics… button to select statistics we will need for the analysis.

The baseline regression

Retain the default checkboxes for Estimates and Model fit to obtain the baseline R², which will be used to determine whether we should use the model with transformations and excluding outliers, or the model with the original form of the variables and all cases.

Mark the Descriptives checkbox to get the number of cases available for the analysis.

Mark the checkbox for the Durbin-Watson statistic, which will be used to test the assumption of independence of errors.

Initial sample size

The initial sample size before excluding outliers and influential cases is 254. With 3 independent variables, the ratio of cases to variables is 84.7 to 1, satisfying both the minimum and preferred sample size for multiple regression.

If the sample size did not initially satisfy the minimum requirement, regression analysis is not appropriate.

R² before transformations or removing outliers

The R² of 0.187 is the benchmark that we will use to evaluate the utility of transformations and the elimination of outliers/influential cases.

Prior to any transformations of variables to satisfy the assumptions of multiple regression or removal of outliers, the proportion of variance in the dependent variable explained by the independent variables (R²) was 18.7%.

The relationship is statistically significant, though we would not stop if it were not significant because the lack of significance may be a consequence of violation of assumptions or the inclusion of outliers and influential cases.

Assumption of independence of errors:the Durbin-Watson statistic

The Durbin-Watson statistic is used to test for the presence of serial correlation among the residuals, i.e., the assumption of independence of errors, which requires that the residuals or errors in prediction do not follow a pattern from case to case.

The value of the Durbin-Watson statistic ranges from 0 to 4. As a general rule of thumb, the residuals are not correlated if the Durbin-Watson statistic is approximately 2, and an acceptable range is 1.50 - 2.50.

The Durbin-Watson statistic for this problem is 1.849 which falls within the acceptable range.

If the Durbin-Watson statistic was not in the acceptable range, we would add a caution to the findings for a violation of regression assumptions.

Normality of dependent variable:how many in family earned money

After evaluating the dependent variable, we examine the normality of each metric variable and linearity of its relationship with the dependent variable.

To test the normality of number of earners in family, run the script: NormalityAssumptionAndTransformations.SBS

First, move the independent variable EARNRS to the list box of variables to test.

Second, click on the OK button to produce the output.

Normality of dependent variable:how many in family earned money

The dependent variable "how many in family earned money" [earnrs] does not satisfy the criteria for a normal distribution.

The skewness (0.742) fell between -1.0 and +1.0, but the kurtosis (1.324) fell outside the range from -1.0 to +1.0.

Normality of dependent variable:how many in family earned money

The logarithmic transformation improves the normality of "how many in family earned money" [earnrs]. In evaluating normality, the skewness (-0.483) and kurtosis (-0.309) were both within the range of acceptable values from -1.0 to +1.0.

The square root transformation also has values of skewness and kurtosis in the acceptable range.

However, by our order of preference for which transformation to use, the logarithm is preferred to the square root or inverse.

Transformation for how many in family earned money

- The logarithmic transformation improves the normality of "how many in family earned money" [earnrs].
- We will substitute the logarithmic transformation of how many in family earned money as the dependent variable in the regression analysis.

Adding a transformed variable

Before testing the assumptions for the independent variables, we need to add the transformation of the dependent variable to the data set.

First, move the variable that we want to transform to the list box of variables to test.

Second, mark the checkbox for the transformation we want to add to the data set, and clear the other checkboxes.

Third, clear the checkbox for Delete transformed variables from the data. This will save the transformed variable.

Fourth, click on the OK button to produce the output.

The transformed variable in the data editor

If we scroll to the extreme right in the data editor, we see that the transformed variable has been added to the data set.

Whenever we add transformed variables to the data set, we should be sure to delete them before starting another analysis.

Normality/linearity of independent variable: age

After evaluating the dependent variable, we examine the normality of each metric variable and linearity of its relationship with the dependent variable.

To test the normality of age, run the script: NormalityAssumptionAndTransformations.SBS

First, move the independent variable AGE to the list box of variables to test.

Second, click on the OK button to produce the output.

Normality/linearity of independent variable: age

In evaluating normality, the skewness (0.595) and kurtosis (-0.351) were both within the range of acceptable values from -1.0 to +1.0.

Normality/linearity of independent variable: age

First, move the transformed dependent variable LOGEARN to the text box for the dependent variable.

To evaluate the linearity of age and the log transformation of number of earners in the family, run the script for the assumption of linearity:

LinearityAssumptionAndTransformations.SBS

Second, move the independent variable, AGE, to the list box for independent variables.

Third, click on the OK button to produce the output.

Normality/linearity of independent variable: age

The evidence of linearity in the relationship between the independent variable "age" [age] and the dependent variable "log transformation of how many in family earned money" [logearn] was the statistical significance of the correlation coefficient (r = -0.493). The probability for the correlation coefficient was <0.001, less than or equal to the level of significance of 0.01. We reject the null hypothesis that r = 0 and conclude that there is a linear relationship between the variables.

The independent variable "age" [age] satisfies the criteria for both the assumption of normality and the assumption of linearity with the dependent variable "log transformation of how many in family earned money" [logearn].

Normality/linearity of independent variable: respondent's socioeconomic index

First, move the independent variable SEI to the list box of variables to test.

To test the normality of respondent's socioeconomic index, run the script:

NormalityAssumptionAndTransformations.SBS

Second, click on the OK button to produce the output.

Normality/linearity of independent variable: respondent's socioeconomic index

The independent variable "respondent's socioeconomic index" [sei] satisfies the criteria for the assumption of normality, but does not satisfy the assumption of linearity with the dependent variable "log transformation of how many in family earned money" [logearn].

In evaluating normality, the skewness (0.585) and kurtosis (-0.862) were both within the range of acceptable values from -1.0 to +1.0.

Normality/linearity of independent variable: respondent's socioeconomic index

First, move the transformed dependent variable LOGEARN to the text box for the dependent variable.

To evaluate the linearity of the relationship between respondent's socioeconomic index and the log transformation of how many in family earned money, run the script for the assumption of linearity:

LinearityAssumptionAndTransformations.SBS

Second, move the independent variable, SEI, to the list box for independent variables.

Third, click on the OK button to produce the output.

Normality/linearity of independent variable: respondent's socioeconomic index

The probability for the correlation coefficient was 0.385, greater than the level of significance of 0.01. We cannot reject the null hypothesis that r = 0, and cannot conclude that there is a linear relationship between the variables.

Since none of the transformations to improve linearity were successful, it is an indication that the problem may be a weak relationship, rather than a curvilinear relationship correctable by using a transformation. A weak relationship is not a violation of the assumption of linearity, and does not require a caution.

Homoscedasticity of independent variable: Sex

First, move the transformed dependent variable LOGEARN to the text box for the dependent variable.

To evaluate the homoscedasticity of the relationship between sex and the log transformation of how many in family earned money, run the script for the assumption of homogeneity of variance:

HomoscedasticityAssumptionAnd Transformations.SBS

Second, move the independent variable, SEX, to the list box for independent variables.

Third, click on the OK button to produce the output.

Homoscedasticity of independent variable: Sex

Based on the Levene Test, the variance in "log transformation of how many in family earned money" [logearn] is homogeneous for the categories of "sex" [sex].

The probability associated with the Levene Statistic (0.767) is greater than the level of significance, so we fail to reject the null hypothesis and conclude that the homoscedasticity assumption is satisfied.

The regression to identify outliers and influential cases

We use the regression procedure to identify univariate outliers, multivariate outliers, and influential cases.

We start with the same dialog we used for the baseline analysis and substitute the transformed variables which we think will improve the analysis.

To run the regression again, select the Regression | Linear command from the Analyze menu.

The regression to identify outliers and influential cases

First, we substitute the logarithmic transformation of earnrs, logearn, into the list of independent variables.

Second, we keep the method of entry to Enter so that all variables will be included in the detection of outliers.

NOTE: we should always use Enter when testing for outliers and influential cases to make sure all variables are included in the determination.

Third, we want to save the calculated values of the outlier statistics to the data set.

Click on the Save… button to specify what we want to save.

Saving the measures of outliers/influential cases

First, mark the checkbox for Studentized residuals in the Residuals panel. Studentized residuals are z-scores computed for a case based on the data for all other cases in the data set.

Fourth, click on the OK button to complete the specifications.

Second, mark the checkbox for Mahalanobis in the Distances panel. This will compute Mahalanobis distances for the set of independent variables.

Third, mark the checkbox for Cook’s in the Distances panel. This will compute Cook’s distances to identify influential cases.

The variables for identifying outliers/influential cases

The variable for identifying univariate outliers for the dependent variable are in a column which SPSS has named sre_1. These are the studentized residuals for the log transformed variables.

The variable for identifying multivariate outliers for the independent variables are in a column which SPSS has named mah_1.

The variable containing Cook’s distances for identifying influential cases has been named coo_1 by SPSS.

Computing the probability for Mahalanobis D²

To compute the probability of D², we will use an SPSS function in a Compute command.

First, select the Compute… command from the Transform menu.

Formula for probability for Mahalanobis D²

First, in the target variable text box, type the name "p_mah_1" as an acronym for the probability of the mah_1, the Mahalanobis D² score.

Second, to complete the specifications for the CDF.CHISQ function, type the name of the variable containing the D² scores, mah_1, followed by a comma, followed by the number of variables used in the calculations, 3.

Since the CDF function (cumulative density function) computes the cumulative probability from the left end of the distribution up through a given value, we subtract it from 1 to obtain the probability in the upper tail of the distribution.

Third, click on the OK button to signal completion of the computer variable dialog.

Univariate outliers

A score on the dependent variable is considered unusual if its studentized residual is bigger than ±3.0.

Multivariate outliers

The combination of scores for the independent variables is an outlier if the probability of the Mahalanobis D² distance score is less than or equal to 0.001.

Influential cases

In addition, a case may have a large influence on the regression analysis, resulting in an analysis that is less representative of the population represented by the sample. The criteria for identifying influential case is a Cook's distance score with a value of 0.0160 or greater.

The criteria for Cook’s distance is:

4 / (n – k – 1) =

4 / (254 – 3 – 1) = 0.0160

Omitting the outliers and influential cases

To omit the outliers and influential cases from the analysis, we select in the cases that are not outliers and are not influential cases.

First, select the Select Cases… command from the Transform menu.

Specifying the condition to omit outliers

First, mark the If condition is satisfied option button to indicate that we will enter a specific condition for including cases.

Second, click on the If… button to specify the criteria for inclusion in the analysis.

The formula for omitting outliers

To eliminate the outliers and influential cases, we request the cases that are not outliers or influential cases.

The formula specifies that we should include cases if the studentized residual (regardless of sign) is less than 3, the probability for Mahalanobis D² is higher than the level of significance of 0.001, and the Cook’s distance value is less than the critical value of 0.0160.

After typing in the formula, click on the Continue button to close the dialog box,

Completing the request for the selection

To complete the request, we click on the OK button.

An omitted outlier and influential case

SPSS identifies the excluded cases by drawing a slash mark through the case number. This omitted case has a large studentized residual, greater than 3.0, as well as a Cook’s distance value that is greater than the critical value, 0.0160.

The outliers and influential cases

Case 20000159 is an influential case (Cook's distance=0.0320) as well as an outlier on the dependent variable (studentized residual=3.13). Case 20000915 is an influential case (Cook's distance=0.0239). Case 20001016 is an influential case (Cook's distance=0.0598) as well as an outlier on the dependent variable (studentized residual=-3.12). Case 20001761 is an influential case (Cook's distance=0.0167). Case 20002587 is an influential case (Cook's distance=0.0264). Case 20002597 is an influential case (Cook's distance=0.0293). There are 6 cases that have a Cook's distance score that is large enough to be considered influential cases.

Running the regression omitting outliers

We run the regression again, without the outliers which we selected out with the Select If command.

Select the Regression | Linear command from the Analyze menu.

Opening the save options dialog

We specify the dependent and independent variables, continuing to substitute any transformed variables required by assumptions.

On our last run, we instructed SPSS to save studentized residuals, Mahalanobis distance, and Cook’s distance. To prevent these values from being calculated again, click on the Save… button.

Clearing the request to save outlier data

First, clear the checkbox for Studentized residuals.

Second, clear the checkbox for Mahalanobis distance.

Third, click on the OK button to complete the specifications.

Third, clear the checkbox form Cook’s distance.

Opening the statistics options dialog

Once we have removed outliers, we need to check the sample size requirement for regression.

Since we will need the descriptive statistics for this, click on the Statistics… button.

Requesting descriptive statistics

First, mark the checkbox for Descriptives.

Third, click on the Continue button to complete the specifications.

Second, mark the checkbox for Collinearity diagnostics to obtain the tolerance values for each independent variable in order to assess multicollinearity.

Requesting the output

Having specified the output needed for the analysis, we click on the OK button to obtain the regression output.

SELECTION OF MODEL FOR INTERPRETATION

Prior to any transformations of variables to satisfy the assumptions of multiple regression and the removal of outliers and influential cases, the proportion of variance in the dependent variable explained by the independent variables (R²) was 18.7%.

After substituting transformed variables and removing outliers and influential cases, the proportion of variance in the dependent variable explained by the independent variables (R²) was 38.4%.

Since the regression analysis using transformations and omitting outliers and influential cases explained at least two percent more variance than the regression analysis with all cases and no transformations, the regression analysis with transformed variables omitting outliers and influential cases was interpreted.

SAMPLE SIZE

The minimum ratio of valid cases to independent variables for multiple regression is 5 to 1. After removing 6 influential cases or outliers, there are 248 valid cases and 3 independent variables.

The ratio of cases to independent variables for this analysis is 82.67 to 1, which satisfies the minimum requirement. In addition, the ratio of 82.67 to 1 satisfies the preferred ratio of 15 to 1.

OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT VARIABLES

The probability of the F statistic (50.759) for the overall regression relationship is <0.001, less than or equal to the level of significance of 0.05. We reject the null hypothesis that there is no relationship between the set of independent variables and the dependent variable (R² = 0). We support the research hypothesis that there is a statistically significant relationship between the set of independent variables and the dependent variable.

We support the research hypothesis that there is a statistically significant relationship between the set of independent variables and the dependent variable.

OVERALL RELATIONSHIP BETWEEN INDEPENDENT AND DEPENDENT VARIABLES

The Multiple R for the relationship between the set of independent variables and the dependent variable is 0.620, which would be characterized as strong using the rule of thumb than a correlation less than or equal to 0.20 is characterized as very weak; greater than 0.20 and less than or equal to 0.40 is weak; greater than 0.40 and less than or equal to 0.60 is moderate; greater than 0.60 and less than or equal to 0.80 is strong; and greater than 0.80 is very strong.

MULTICOLLINEARITY

Multicollinearity occurs when one independent variable is so strongly correlated with one or more other variables that its relationship to the dependent variable is likely to be misinterpreted. Its potential unique contribution to explaining the dependent variable is minimized by its strong relationship to other independent variables. Multicollinearity is indicated when the tolerance value for an independent variable is less than 0.10.

The tolerance values for all of the independent variables are larger than 0.10. Multicollinearity is not a problem in this regression analysis.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 1

For the independent variable age, the probability of the t statistic (-12.237) for the b coefficient is <0.001 which is less than or equal to the level of significance of 0.05. We reject the null hypothesis that the slope associated with age is equal to zero (b = 0) and conclude that there is a statistically significant relationship between age and log transformation of how many in family earned money.

The b coefficient associated with age (-0.007) is negative, indicating an inverse relationship in which higher numeric values for age are associated with lower numeric values for log transformation of how many in family earned money. Therefore, the negative value of b implies that survey respondents who were older had fewer family members earning money.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 2

For the independent variable sex, the probability of the t statistic (1.284) for the b coefficient is 0.200 which is greater than the level of significance of 0.05.

We fail to reject the null hypothesis that the slope associated with sex is equal to zero (b = 0) and conclude that there is not a statistically significant relationship between sex and log transformation of how many in family earned money.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 3

For the independent variable respondent's socioeconomic index, the probability of the t statistic (0.354) for the b coefficient is 0.724 which is greater than the level of significance of 0.05.

We fail to reject the null hypothesis that the slope associated with respondent's socioeconomic index is equal to zero (b = 0) and conclude that there is not a statistically significant relationship between respondent's socioeconomic index and log transformation of how many in family earned money.

Validation analysis:set the random number seed

To set the random number seed, select the Random Number Seed… command from the Transform menu.

Set the random number seed

First, click on the Set seed to option button to activate the text box.

Second, type in the random seed stated in the problem.

Third, click on the OK button to complete the dialog box.

Note that SPSS does not provide you with any feedback about the change.

Validation analysis:compute the split variable

To enter the formula for the variable that will split the sample in two parts, click on the Compute… command.

The formula for the split variable

First, type the name for the new variable, split, into the Target Variable text box.

Second, the formula for the value of split is shown in the text box.

The uniform(1) function generates a random decimal number between 0 and 1. The random number is compared to the value 0.50.

If the random number is less than or equal to 0.50, the value of the formula will be 1, the SPSS numeric equivalent to true. If the random number is larger than 0.50, the formula will return a 0, the SPSS numeric equivalent to false.

Third, click on the OK button to complete the dialog box.

The split variable in the data editor

In the data editor, the split variable shows a random pattern of zero’s and one’s.

To select half of the sample for each validation analysis, we will first select the cases where split = 0, then select the cases where split = 1.

Repeat the regression with first validation sample

To repeat the multiple regression analysis for the first validation sample, select Linear Regression from the Dialog Recall tool button.

Using "split" as the selection variable

First, scroll down the list of variables and highlight the variable split.

Second, click on the right arrow button to move the split variable to the Selection Variable text box.

Setting the value of split to select cases

When the variable named split is moved to the Selection Variable text box, SPSS adds "=?" after the name to prompt up to enter a specific value for split.

Click on the Rule… button to enter a value for split.

Completing the value selection

First, type the value for the first half of the sample, 0, into the Value text box.

Second, click on the Continue button to complete the value entry.

Requesting output for the first validation sample

Click on the OK button to request the output.

When the value entry dialog box is closed, SPSS adds the value we entered after the equal sign. This specification now tells SPSS to include in the analysis only those cases that have a value of 0 for the split variable.

Since the validation analysis requires us to compare the results of the analysis using the two split sample, we will request the output for the second sample before doing any comparison.

Repeat the regression with second validation sample

To repeat the multiple regression analysis for the second validation sample, select Linear Regression from the Dialog Recall tool button.

Setting the value of split to select cases

Since the split variable is already in the Selection Variable text box, we only need to change its value.

Click on the Rule… button to enter a different value for split.

Completing the value selection

First, type the value for the first half of the sample, 1, into the Value text box.

Second, click on the Continue button to complete the value entry.

Requesting output for the second validation sample

Click on the OK button to request the output.

When the value entry dialog box is closed, SPSS adds the value we entered after the equal sign. This specification now tells SPSS to include in the analysis only those cases that have a value of 1 for the split variable.

SPLIT-SAMPLE VALIDATION - 1

In both of the split-sample validation analyses, the relationship between the independent variables and the dependent variable was statistically significant.

In the first validation, the probability for the F statistic testing overall relationship was <0.001.

For the second validation analysis, the probability for the F statistic testing overall relationship was <0.001.

Thus far, the validation verifies the existence of the relationship between the dependent variable and the independent variables.

SPLIT-SAMPLE VALIDATION - 2

The total proportion of variance in the relationship utilizing the full data set was 38.4% compared to 40.0% for the first split sample validation and 36.5% for the second split sample validation.

In both of the split-sample validation analyses, the total proportion of variance in the dependent variable explained by the independent variables was within 5% of the variance explained in the model using the full data set (38.4%).

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 1

The relationship between "age" [age] and "log transformation of how many in family earned money" [logearn] was statistically significant for the model using the full data set (p<0.001). Similarly, the relationships in both of the validation analyses were statistically significant.

In the first validation analysis, the probability for the test of relationship between "age" [age] and "log transformation of how many in family earned money" [logearn] was <0.001, which was less than or equal to the level of significance of 0.05 and statistically significant.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 2

In the second validation analysis, the probability for the test of relationship between "age" [age] and "log transformation of how many in family earned money" [logearn] was <0.001, which was less than or equal to the level of significance of 0.05 and statistically significant.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 3

The relationship between "respondent's socioeconomic index" [sei] and "log transformation of how many in family earned money" [logearn] was not statistically significant for the model using the full data set (p=0.724). Similarly, the relationships in both of the validation analyses were not statistically significant.

In the first validation analysis, the probability for the test of relationship between "respondent's socioeconomic index" [sei] and "log transformation of how many in family earned money" [logearn] was 0.601, which was greater than the level of significance of 0.05 and not statistically significant.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 4

In the second validation analysis, the probability for the test of relationship between "respondent's socioeconomic index" [sei] and "log transformation of how many in family earned money" [logearn] was 0.274, which was greater than the level of significance of 0.05 and not statistically significant.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 5

The relationship between "sex" [sex] and "log transformation of how many in family earned money" [logearn] was not statistically significant for the model using the full data set (p=0.200). Similarly, the relationships in both of the validation analyses were not statistically significant.

In the first validation analysis, the probability for the test of relationship between "sex" [sex] and "log transformation of how many in family earned money" [logearn] was 0.410, which was greater than the level of significance of 0.05 and not statistically significant.

RELATIONSHIP OF INDIVIDUAL INDEPENDENT VARIABLES TO DEPENDENT VARIABLE - 6

In the second validation analysis, the probability for the test of relationship between "sex" [sex] and "log transformation of how many in family earned money" [logearn] was 0.366, which was greater than the level of significance of 0.05 and not statistically significant.

The split sample validation supports the findings of the regression analysis using the full data set.

The answer to the original question is true.

Table of validation results: standard regression

It may be helpful to create a table for our validation results and fill in its cells as we complete the analysis.

The split sample validation supports the findings of the regression analysis using the full data set.

Answering the problem question - 1In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

Survey respondents who were older had fewer family members earning money. The variables sex and respondent's socioeconomic index did not have a relationship to how many in family earned money.

We have found that there is a statistically significant relationship between the set of IVs and the DV (p<0.001), and the Multiple R was 0.620, which would be characterized as a strong relationship.

1. True

2. True with caution

3. False

4. Inappropriate application of a statistic

Answering the problem question - 2In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

Survey respondents who were older had fewer family members earning money. The variables sex and respondent's socioeconomic index did not have a relationship to how many in family earned money.

The b coefficient associated with age was statistically significant

(p< 0.001), so there was an individual relationship to interpret.

The b coefficient (-0.007) was negative, indicating an inverse relationship in which higher numeric values for age are associated with lower numeric values for log transformation of how many in family earned money.

Therefore, the negative value of b implies that survey respondents who were older had fewer family members earning money.

1. True

2. True with caution

3. False

4. Inappropriate application of a statistic

Answering the problem question - 3In the dataset GSS2000.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that there is no problem with missing data. Use a level of significance of 0.05 for the regression analysis. Use a level of significance of 0.01 for evaluating assumptions. Use 0.0160 as the criteria for identifying influential cases. Validate the results of your regression analysis by splitting the sample in two, using 788035 as the random number seed.

The variables "age" [age], "sex" [sex], and "respondent's socioeconomic index" [sei] have a strong relationship to the variable "how many in family earned money" [earnrs].

Survey respondents who were older had fewer family members earning money. The variables sex and respondent's socioeconomic index did not have a relationship to how many in family earned money.

For the independent variable sex, the probability of the t statistic (1.284) for the b coefficient is 0.200 which is greater than the level of significance of 0.05. Sex did not have a relationship to the number of persons in the family earning money.

For the independent variable respondent's socioeconomic index, the probability of the t statistic (0.354) for the b coefficient is 0.724 which is greater than the level of significance of 0.05. Socioeconomic status did not have a relationship to the number of persons in the family earning money.

1. True

2. True with caution

3. False

4. Inappropriate application of a statistic

The answer to the question is true.

Yes

Yes

Steps in regression analysis: running the baseline modelThe following is a guide to the decision process for answering

problems about the complete regression analysis:

Dependent variable metric?

Independent variables metric or dichotomous?

Inappropriate application of a statistic

No

Yes

Ratio of cases to independent variables at least 5 to 1?

Inappropriate application of a statistic

Run baseline regression, using method for including variables identified in the research question.

Record R² for evaluation of transformations and removal of outliers and influential cases.

Record Durbin-Watson statistic for assumption of independence of errors.

No

Yes

Yes

Steps in regression analysis: evaluating assumptions - 1Try:

1. Logarithmic transformation

2. Square root transformation

3. Inverse transformation

If unsuccessful, add caution for violation of regression assumptions

Is the dependent variable normally distributed?

Try:

1. Logarithmic transformation

2. Square root transformation

(3. Square transformation)

4. Inverse transformation

If unsuccessful, add caution for violation of regression assumptions

Metric IV’s normally distributed and linearly related to DV?

No

Yes

Yes

Steps in regression analysis: evaluating assumptions - 2DV is homoscedastic for categories of dichotomous IV’s?

Add caution for violation of regression assumptions

Residuals are independent,

Durbin-Watson between 1.5 and 2.5?

Add caution for violation of regression assumptions

No

No

Steps in regression analysis: evaluating outliersRequest statistics for detecting outliers and influential cases by running standard multiple regression using Enter method to include all variables and substituting transformed variables.

Univariate outliers (DV), multivariate outliers (IVs), or influential cases?

Remove outliers and influential cases from data set

Ratio of cases to independent variables at least 5 to 1?

Restore outliers and influential cases to data set, add caution to findings

Yes

Yes

Yes

No

Steps in regression analysis: picking regression model for interpretationWere transformed variables substituted, or outliers and influential cases omitted?

Evaluate impact of transformations and removal of outliers by running regression again, using method for including variables identified in the research question.

R² for evaluated regression greater than R² for baseline regression by 2% or more?

Pick regression with transformations and omitting outliers for interpretation

Pick baseline regression

for interpretation

No

Yes

Yes

Steps in regression analysis: overall relationship is interpretableProbability of ANOVA test of regression less than/equal to level of significance?

False

Tolerance for all IV’s greater than 0.10, indicating no multicollinearity?

False

Yes

Yes

Steps in regression analysis: validation - 1Enough valid cases to split sample and keep 5 to 1 ratio of cases/variables?

No

Yes

- Set the first random seed and compute the split1 variable
- Re-run regression with split1 = 1
- Set the second random seed and compute the split2 variable
- Re-run regression with split2 = 1

- Set the random seed and compute the split variable
- Re-run regression with split = 0
- Re-run regression with split = 1

Probability of ANOVA test <= level of significance for both validation analyses?

False

No

No

Yes

Yes

Yes

Steps in regression analysis: validation - 2R² for both validations within 5% of R² for analysis of full data set?

False

Change in R² statistically significant in both validation analyses? (Hierarchical only)

False

Pattern of significance for independent variables in both validations matches pattern for full data set?

False

No

No

Yes

Yes

Yes

Yes

Yes

Steps in regression analysis: answering the questionSatisfies ratio for preferred sample size: 15 to 1

(stepwise: 50 to 1)

True with caution

DV is interval level and IVs are interval level or dichotomous?

True with caution

Assumptions not violated? Outliers/influential cases excluded from interpretation?

True with caution

True

Interpreting b coefficients when the dependent variable is transformed - 1

For the independent variable age, the probability of the t statistic (-12.237) for the b coefficient is <0.001 which is less than or equal to the level of significance of 0.05. We reject the null hypothesis that the slope associated with age is equal to zero (b = 0) and conclude that there is a statistically significant relationship between age and log transformation of how many in family earned money.

The b coefficient associated with age (-0.007) is negative, indicating an inverse relationship in which higher numeric values for age are associated with lower numeric values for log transformation of how many in family earned money. Therefore, the negative value of b implies that survey respondents who were older had fewer family members earning money.

Interpreting b coefficients when the dependent variable is transformed - 2

If we not to interpret a specific change in number of earners for some amount of change in age, we will need to find the answer and convert it from log units to decimal units. We can use Microsoft Excel to calculate the answer.

In the worksheet, I have entered the b coefficient from the SPSS output (-0.007) in row 1.

In row 2, I have entered different ages, e.g. 20, 30, 40, and 50.

Interpreting b coefficients when the dependent variable is transformed - 3

On row 3, we multiply the value for the independent variable age by the b coefficient, which measures the contribution to the dependent variable in log units.

On row 4, we reverse the log transform back to decimal units by raising the number 10 the value on row 3.

The caret symbol is used by Excel for raise to a power, so 10^-0.14 is 10 raised to the -0.14 power, or 0.7244.

Interpreting b coefficients when the dependent variable is transformed - 4

Based on our table, ia respondent were age 20 contributes 0.7244 to the number of earners in the family. If a respondent were 30, rather than 20, the contribution to number of earns would be 0.6166, a decrease of -0.1078. Thus, increasing age has a negative effect on the number of earners.

Note that as we go up in increments of 10, the difference between increments is decreasing. The logarithmic scale is not linear, requiring us to compute the change for any specific interval of interest.

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