## UP504 • Multiple Regression

last updated: Tuesday, January 15, 2008

BIG VERSION

# Assignment:

Assignment One requires you to develop a multiple regression model. The assignment is described on a separate page.

# Note:

Regression is a powerful, complex tool with MANY variations and requirements.  Please refer to the class readings for a comprehensive discussion.  These notes are merely to supplement the required readings.

# What is regression?   Depending on your goal and priorities and perspective, it is:

• a powerful statistical method to test the strength and significance of relationships between variables
• the search for (usually linear) patterns in the mess of scatterplots
• a technique to develop an explanatory model (what explains the variation in the dependent variable Y?)
• a technique to develop a predictive model (given specific values for the independent variables, what is our best estimate of Y?)
• a way to isolate the influence and significance of one variable (x) on another (y) by controlling for all the other independent variables (e.g., to isolate the role of race or gender on income levels controlling for education, years of experience, etc.)
• the hardest, most complex technique learned in a first-semester statistics course
• the hunt for a high R-square (though this is NOT necessarily a legitimate goal for regression)
• based on the assumption that there is some common, underlying relationship between the variables that can be isolated and measured by controlling for all the other variables.

# Lecture Presentation: Example using Regression to Test/Evaluate Policies to lower Vacancy Rates in Public Housing Projects (Jan 9):

download pdf file [one page handout setting up the problem -- see the ppt presentation above for the answer]

# Handout (Jan 14):

World95.sav regression SPSS runs (6 page pdf file: using country-level data to estimate fertility rates)

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# The main goals of learning for this section on regression:

1. how to use regression to address research questions
2. how to use regression equations for predictions
3. using multiple regression to see the unique influence of individual variables
4. knowing when a relationship is significant.
5. understanding the role of linearity, multicollinearity, residuals, outliers.
6. know when regression results might be misleading

# Key Terms include:

 dependent variable (explained) independent variable (explanatory variable) degrees of freedom linear and nonlinear error residuals sum of squares OLS - Ordinary Least Squares multicollinearity dummy variable (a categorical variable coded as 0 or 1 so that it can act as an independent variable)

Lewis-Beck, Michael S. 1980. Applied Regression: An Introduction. Newbury Park, CA: Sage.

# The Role of b, Beta, t, R-square and F:

 each individual independent variable the model as a whole strength of the relationship b (partial regression coefficient);   or for better comparison between variables, use:  Beta weights (standardized b) R 2 (for models with many variables, also look at the adjusted R2) statistical significance t score F score

# These are therefore desirable features of a regression model

 each individual independent variable the model as a whole strength of the relationship large Beta weights (their absolute values)  (though if the t, R2 and F are all "ok" then don't worry directly about the value of b and Beta.) a high R2 (closer to 1 than 0)  that is, most of the variation in Y explained. statistical significance a high and thus significant t score (generally, |t| > 2.  (remember:  ALL variables in a model need to be stat. significant) a high and thus significant F (see the F table, but generally above about 4 to be sign. at .05 level)

In addition, error terms have a constant variance, no or only a few outliers, error terms normally distributed, little multicollinearity (independent variables that are highly correlated), etc.

# Overview of regression statistics and their use:

 statistic formula Questions we ask b As x increases by one unit, how much does Y increase?  (use to construct the regression equation).  known as the regression coefficient, or in multiple regression as the "partial regression coefficient" Beta weights standardized regression coefficient. As x increases by one standard deviation, how much does Y increase (in standard deviations)?  Useful to compare the relative explanatory power of different independent variables (especially when ind. variables have different measurement scales). Beta weights can be interpreted like partial r (correlation coefficients). a a = y - bx What is the y-intercept?  (That is, when x = 0, what is y?).   Sometimes this value has real meaning, sometimes not. Generally, when the y-intercept falls within the range of data values, it will be more meaningful than when it falls outside the range of data values. t t = b / std. error What is the statistical significance of the relationship between this independent variable and the dependent variable (controlling for the other variables in the model)?  (SPSS calculates the probability of this t score being due to just random chance, labeled "Sig" for "Significance", where the number represents the chances out of 1 that the measured difference might be just due to random chance.)   Generally we consider variables with Sig values < .05 to be statistically significant. R2 What percent of the total variation in the dependent variable is explained by the independent variables in the model?  R-square = explained (or regression) sum of squares / total sum of squares.     or   R-square = RSS / TSS =   1 - (SSE / TSS) F or What is the statistical significance of the model as a whole?  (SPSS calculates a significance level for this, similar to that for the t scores.) k . the number of independent variables n . the number of cases degrees of freedom for regression (explained):  k for residual (unexplained):  n - k -1 total:  n - 1 Not itself interpreted, but used to calculate the other statistics; defined (Blalock, 205): "equal to the number of quantities that are unknown minus the number of independent equations linking these unknowns."   that extra one degree lost is due to the dependent variable

Some terms

F-Score
The F-score from the ANOVA table (Analysis of Variance) allows one to determine the probability of getting these regression results if there was no difference in the population as a whole. What is a significant F-score? It depends on the degrees of freedom (both the number of independent variables, k, and the total number of cases, n, or more precisely, n- k-1). See an F-table (in the back of stat books). For example, with 4 independent variables and 30 cases, F is significant at the p=.05 level when F>2.76. With 4 variables and 125 cases, the threshold is F>2.45. (You will generally find that your regression models will always have stat. significant F-scores; it is harder to develop a powerful, meaningful model where all of the variables have stat. significant t-scores.)

Beta weights
Beta weights are adjusted partial slopes, or standardized B's. [see Lewis-Beck, p. 64]  To calculate, multiply the b by the standard deviation of the dependent variable (x), and divide by the standard deviation of the independent variable (y). Beta weights are useful for comparing the relative importance of each independent variable. Compare the absolute values of the beta weights. (For example: if your model has two independent variables -- the first with a Beta weight of -0.566 and the second with a Beta weight of 0.231 -- the first variable is a more powerful explanatory variable in the model.)

The Adjusted R-Square takes into account not only how much of the variation is explained, but also the impact of the degrees of freedom. It "adjusts" for the number of variables use.   That is, look at the adjusted R- Square to see how adding another variable to the model both increases the explained variance but also lowers the degrees of freedom.
Adjusted R2 = 1- (1 - R2 )((n - 1)/(n - k - 1)).  As the number of variables in the model increases, the gap between the R-square and the adjusted R-square will increase.   This serves as a disincentive to simply throwing in a huge number of variables into the model to increase the R-square.

Ordinary Least Squares (OLS)
In regression the goal is to find the best fitting equation that links the independent variables with the dependent variable.  This is one that minimizes the error of prediction.   How is this error minimized? A simple approach is to simply minimize the sum of squares (i.e., "least squares") of the vertical distances between the estimate line (estimate) and the actual value of y.   (This is SSE - the sum of the square of errors). There are numerous other methods (and advantages of each), such as weighted least squares (WLS), 2-Stage Least Squares (2SLS), etc.

Thus: OLS is a method that estimates an equation for the regression line by minimizing the sum of the square of differences between the actual value of each case and its predicted value:

Why might an R-Square be less than 1.00?

• underdetermined model (need more variables)
• nonlinear relationships
• measurement error
• sampling error
• not fully predictable/explainable even with all data available; there is a certain amount of unexplainable chaos/static/randomness in the universe (which may be reassuring)
• the unit of analysis is too aggregated (e.g., you are predicting mean housing values for a city -- you might get better results with predicting individual housing prices, or neighborhood housing prices).

Is an R-Square < 1.00 Good or bad?
This is both a statistical and a philosophical question;
It is quite rare, especially in the social sciences, to get an R-square that is really high (e.g., 98%).
The goal is NOT to get the highest R-square per se.   Instead, the goal is to develop a model that is both statistically and theoretically  sound, creating the best fit with existing data.

Do you want just the best fit, or a model that theoretically/conceptually makes sense?
Yes, you might get a good fit with nonsensical explanatory variables.  But, this opens you to spurious/intervening relationships. THEREFORE: hard to use model for explanation.

# What is needed to run a regression

1. at least two variables (both interval)
2. enough cases to be statistically significant
3. some basic computations; can do by hand, with a calculator, with Excel by calculations, or EXCEL regression function; of a dedicated stat program, such as SPSS, SAS, Systat, etc.
4. an understanding of the requirements of regression so that you don't violate some basic statistical rules.

# From Bivariate to Multiple regression: what changes?

• potentially more explanatory power with more variables.
• the ability to control for other variables: and one sees the interaction of the various explanatory variables. partial correlations and multicollinearity.
• harder to visualize drawing a line through three+ n-dimensional space.
• the R is no longer simply the square of the correlation statistic r.

Regression Assumptions include:

• linear relationship
• error terms have a constant variance
• no or only a few outliers (always nice to be able to explain why)
• error terms normally distributed
• error terms independent
• little multicollinearity (independent variables that are highly correlated) see Blalock, 485. PROBLEMS: more ambiguity in causal interpretations; partial correlations and slope estimates become more sensitive to sampling (deviations from a representative sample) and measurement (problems with our measures) errors.
• SO: if one has lots of multicollinearity, then one needs BOTH large samples and accurate measurement.
• we will see examples of this in the fertility example: e.g., variables that are all affected by the level of development (literacy, wealth, life expectancy), and culture.

# Residual Plots and Regression Assumptions

Recall that there are three basic assumptions about the random deviations (errors), : the random deviations are independent, normally distributed, and have a constant variance. In simple linear regression, we also assume that Y and X are linearly related. We shall consider the use of residual plots for examining the following types of departures from the assumed model.

1.  The regression function is not linear.

2.  The error terms do not have a constant variance.

3.  The model fits all but one or a few outlying observations.

4.  The error terms are not normally distributed.

5.  The error terms are not independent.

>>> see Lewis-Beck (Applied Regression), page 26, for a good discussion of these assumptions <<<

The common graphical tools for assumption checking includes:

1.  Residual Plot- scatter plot the residuals against X or the fitted value.

2.  Absolute Residual Plot- scatter plot the absolute values of the residuals against X or the fitted value.

3.  Normal Probability Plot of the Residuals.

4.  Time Series Plot of the Residuals - scatter plot the residuals against time or index.

5.  The time series plot of the residuals are strongly recommended whenever data are obtained in a time sequence. The purpose is to see if there is any correlation between the error terms over time (the error terms are not independent). When the error terms are independent, we expect the residuals to fluctuate in a more or less random pattern around the base line 0.

Further Issues:
1. non-linear transformations

2. dummy variables

3. what to do with ordinal variables

4. WLS - weighted least squares.

5. handling interaction between independent variables, that is, multiplicative relationships. (not the assumption with OLS that the influences of ind. variables are additive). e.g., in a JTPA program, to increase ones wage, one may need BOTH job training and additional attributes): one alone won't do as much. That is, each alone raises wages by \$1000/year, but together the effect is +\$7,000.

{this is handling interaction as crossproducts} see Blalock, 492.

Other Techniques
what to do when the dependent variable is not an interval variable? logit, probit, maximum likelihood, etc.  (see statistics books)