Winter, 2010 Tuesday, March 9Stat 217 – Day 27Topics in RegressionExample 1: Can we predict later intelligence based on when the child first speaks? A study of cognitivedevelopment in young children recorded the age (in months) at which each of 21 children spoke their first word and their Gesell Adaptive Score, the result of an aptitude test taken much later. The data can be found in Gesell.mtw. (In Blackboard, Course Materials > Stat 217 Data files.)(a) Produce a scatterplot of the relationship between age first spoke and Gesell by choosing Graph > Scatterplot, and then choose the “With Regression” option to overlay the regression line on the graph. Specify Gesell score (C3) as the Y variable and age at first speaking (C2) as the X variable. Does this graph reveal an association between age when a child speaks and later Gesell scores? Describe the nature of the relationship in context.Note: If you mouse over the blue line and wait, it will display the regression equation.(b) Which child lies further from the regression line? (ID numbers are in column 1.)Def: An observation that has a large residual (y – ) is considered an outlier in the regression model. Observations above the line have positive residuals and points below the line have negative residuals.(c) You will also notice another “unusual observation” – a child that didn’t speak until 42 months (case 18). In the Data window, put an * for that child’s Gesell value. When you hit return the scatterplot should update. (If it doesn’t, right click on the scatterplot and select Update Graph Automatically.) Mouse over the blue line, is it much different from before?(d) Similarly, the child that didn’t speak until 26 months (case 2) also stands out. Put an * for that child’s Gesell score and look at the updated graph. Would your description of the overall relationship bethe same as in (a)? How might you modify your description of the overall nature of the relationship in (a)?Def: An observation is considered influential if removing it from the dataset substantially changes the least squares regression equation. Typically, observations that have extreme explanatory (x) values have more potential to be influential. Note: Influential observations often have small residuals as they pull the least-squares regression line close to them.Winter, 2010 Tuesday, March 9Example 2: Forced expiratory volume (FEV, the maximum volume of air that can forcibly blow out in the first second) is a measure of a person’s lung functioning. It is believed the smokers will have lower FEV values indicating weaker lung functioning. Open the Minitab worksheet FEV.mtw. The FEV values are in column 3 and the smoking status (0 = nonsmoker, 1 = smoker) is in column 6.(a) Perform a two-sample t-test to see whether there is a statistically significant difference in the mean FEV scores for smokers and non-smokers by choosing Stat > Basic Statistics > 2-Sample t. Choose the “Samples in one column” option and specify C3 in the Samples box and C6 in the Subscripts box. Press OK. Find the test statistic and p-value in Minitab’s output. Report these values:(b) Is the difference statistically significant? Is the difference in the expected direction (smokers have lower FEV values on average)?(c) The beauty of regression is pretty much all the inference procedures we have learned this quarter can actually be analyzed using regression models! Choose Stat > Regression > Regression, and specify FEV (C3) in the Response box and smoke (C6) in the Predictors box. We can do this since we coded smoking numerically. Scroll way up in the Session window, what do you notice about the test statistic and p-value for the slope coefficient compared to those values from (a)?In fact, the slope coefficient (.711) is exactly the difference in the two group means. As smoking status (our explanatory variable), increases by 1 (switches from 0=nonsmoker to 1=smoker), the change in the predicted FEV (our response variable) is .711!(d) But this is still a positive number, as smokers have a higher FEV average. Let’s see why we are getting this surprising result. Choose Graph > Scatterplot, and then select the With Regression and Groups option. Specify FEV as the Y variable and Age (C2) as the X variable. Click in the Categorical variables box and specify the smoking variable (C6). Press OK. What do you notice about the relationship between FEV and age? Does the direction of this relationship make sense?(e) Pick an age that has smokers and non-smokers, say 15 years. What do you notice about the predictedFEV for non-smokers (black) compared to smokers (red) – which is higher? Does this make sense?(f) Return to Stat > Regression > Regression and put both smoke and age into the Predictors box. You are now running “Multiple Regression” as you can put any number of explanatory variables into the model to help you predict the response variable. The coefficient of smoke is now interpreted as the predicted change in the response variable when the explanatory variable increases by 1 holding all othervariables constant (i.e., after adjusting for the other variables), that is comparing people of the same age.Is the coefficient of smoke positive of
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