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Advanced Statistical Methods: Continuous Variables http://statisticalmethods.wordpress.com. Multiple Regression – Part I firstname.lastname@example.org. The Multiple Regression Model Ŷ = a + b 1 X 1 + b 2 X 2 + ... + b i X i
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Multiple Regression – Part I
Ŷ = a + b1X1 + b2X2 + ... + biXi
- this equationrepresents the best prediction of a DV from several continuous (or dummy) IVs; i.e. itminimizes the squared differences btw. Y and Ŷ least square regression
Goal: arrive at a set of regression coefficients (bs) for the IVs that bring Ŷs as close as possible to Ys values
a= the estimated value of Y when all independent (exploratory) variables are zero (X1,…i = 0).
bimeasures the partial effect of Xi on Y;
= effect of one-unit increase in Xi, holding all other independent variables constant.
The estimated parameters b1, b2, ..., bi are partial regression coefficients; they are different from regression coefficients for bi-variate relationships between Y and each exploratory variable.
(3) Sample size
Multivariate regression also allows for non-linear relationships, by redefining the IV(s): squaring, cubing, .. of the original IV
1. Cases-to-IVs Ratio
Rule of thumb: N>= 50 + 8*m for testing the multiple correlation;
N>=104 + m for testing individual predictors,
where m = no. of IVs
Need higher case-to-IVs ratio when:
2. Screening for outliers among the DV and the IVs
- too highly correlated IVs are put in the same regression model
4.a. Multivariate Normality
For grouped data: assumption pertains to the sampling distribution of means of variables;
Central Limit Theory: with sufficiently large sample size, sampling distributions are normally distributed regardless of the distribution of the variables
What to look for (in ungrouped data):
Shape of distribution: skewness & kurtosis. Frequency histograms; expected normal probability plots; detrend expected normal probability plots
Heteroskedasticity = caused by:
4.a. Errors of prediction are normally distributed around each & every Ŷ
4.b. Residuals have straight line relationship with Ŷs
- If genuine curvilinear relation btw. an IV and the DV, include a square of the IV in the model
4.c. The variance of the residuals about Ŷs is ~the same for all predicted scores (assumption of homoskedasticity)
- heteroskedasticity may occur when:
- some of the variables are skewed, and others are not;
may consider transforming the variable(s)
- one IV interacts with another variable that is not part of the equation
5. Errors of prediction are independent of one another
Durbin-Watson statistic = measure of autocorrelation of errors over the sequence of cases; if significant it indicates non-independence of errors
Standard multiple regression
Sequential (hierarchical) regression
Statistical (stepwise) regression
R² = a + b + c + d + e
R²= the squared multiple correlation; it is
the proportion of variation in the DV that is
predictable from the best linear combination of the IVs
(i.e. coefficient of determination).
R = correlation between the observed and predicted Y values (R = ryŶ )
Adjusted R2 = modification of R2 that adjusts for the number of terms in a model. R2 always increases when a new term is added to a model, but adjusted R2 increases only if the new term improves the model more than would be expected by chance.
(b & d) contribute
to R² but are not assigned
to any of the individual IVs
Table 1: Regression of (DV) Assessment of Socialism in 2003 on (IVs) Social Status, controlling for Gender and Age
**p <0.001; *p < 0.05;
Interpretation of beta (standardized) coefficients: for a one standard deviation unit increase in X, we get a Beta standard deviation change in Y;
Since variables are transformed into z-scores (i.e. standradized), we can assess their relative impact on the DV (assuming they are uncorrelated with each other)
- researcher specifies the order in which IVs are added to the equation;
X1 gets credit for a and b;
X2 for c and d;
X3 for e.
IVs can be added one at a time, or in blocks
The Regression SUM of SQUARES, SS(regression) = SS(total) + SS(residual)SSregression = Sum (Ŷ – Ybar)² = portion of variation in Y explained by the use of the IVs as predictors; SStotal = Sum (Y- Ybar)²SSresidual = Sum (Y- Ŷ)² - the squared sum of errors in predictionsR² = SSreg/SStotal
The Regression MEAN SQUARE : MSS(regression) = SS(regression) / df, df = k where k = no. of variables
The MEAN square residual (error): MSS(residual) = SS(residual) / df, df= n - (k + 1) where n = no. of cases and k= no. of variables.
The null hypothesis for the regression model:
Ho: b1 = b2 = … = bk = 0
The sampling distribution of this statistic is the F-distribution
The Null Hypothesis for individual IVs
The test of H0: bi = 0 evaluates whether Y and X are statistically dependent, ignoring other variables.
We use the t statistic
σB where σB is a standard error of B
n - 2
In partial correlation (pr), the contribution of the other IVs is taken out of both the IV and the DV;
In semi-partial correlation (sr), the contribution of the other IVs is taken out of only the IV (squared) sr shows the unique contribution of the IV to the total variance of the DV
In standard multiple regression, sr² = the unique contribution of the IV to R²in that set of IVs
(for an IV, sr² = the amount by which R² is reduced, if that IV is deleted from the equation)
If IVs are correlated: usually, sum of sri² < R²
Sequential regression: sri² = amount of variance added to R² by each IV at the point that it is added to the model
In SPSS output sri² is „R² Change” for each IV in „Model Summary” Table