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Regression Analysis

Regression Analysis

(1)Objectives of regression analysis

Regression analysis is used when we want to predict a continuous dependent variable from a number of independent variables. If the dependent variable is dichotomous, then logistic regression should be used. (If the split between the two levels of the dependent variable is close to 50-50, then both logistic and linear regression will end up giving we similar results.) The independent variables used in regression can be either continuous or dichotomous. Independent variables with more than two levels can also be used in regression analyses, but they first must be converted into variables that have only two levels. This is called dummy coding and will be discussed later. Usually, regression analysis is used with naturally-occurring variables, as opposed to experimentally manipulated variables, although we can use regression with experimentally manipulated variables. One point to keep in mind with regression analysis is that causal relationships among the variables cannot be determined. While the terminology is such that we say that X "predicts" Y, we cannot say that X causes Y.

We can also construct a normal probability plot. In this plot, the actual scores are ranked and sorted, and an expected normal value is computed and compared with an actual normal value for each case. The expected normal value is the position a case with that rank holds in a normal distribution. The normal value is the position it holds in the actual distribution. Basically, we would like to see our actual values lining up along the diagonal that goes from lower left to upper right. This plot also shows that age is normally distributed.( Howell, 2002)

We can also test for normality within the regression analysis by looking at a plot of the "residuals." Residuals are the difference between obtained and predicted DV scores. (Residuals will be explained in more detail in a later section.) If the data are normally distributed, then residuals should be normally distributed around each predicted DV score. If the data (and the residuals) are normally distributed, the residuals scatterplot will show the majority of residuals at the center of the plot for each value of the predicted score, with some residuals trailing off symmetrically from the center. We might want to do the residual plot before graphing each variable separately because if this residuals plot looks good, then we don't need to do the separate plots. Below is a residual plot of a regression where age of patient and time (in months since diagnosis) are used to predict breast tumor size. These data are not perfectly normally distributed in that the residuals about the zero line appear slightly more spread out than those below the zero line. Nevertheless, they do appear to be fairly normally distributed.

In addition to a graphic examination of the data, we can also statistically examine the data's normality. Specifically, statistical programs such as SPSS will calculate the skewness and kurtosis for each variable; an extreme value for either one would tell we that the data are not ...

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