Example: Data from the Child Health and Development Study

Transcription

1 Example: Data from the Child Health and Development Study Can we use linear regression to examine how well length of gesta:onal period predicts birth weight? First look at the Does a linear rela:onship seem to exist? birth weight (in ounces) length of gestation period (days)

2 R output: Call: lm(formula = birthweight ~ gesta:on) Residuals: Min 1Q Median 3Q Max Coefficients: Es:mate Std. Error t value Pr(> t ) (Intercept) gesta:on <2e- 16 *** Signif. codes: 0 *** ** 0.01 * Residual standard error: on 1221 degrees of freedom (13 observa:ons deleted due to missingness) Mul:ple R- squared: , Adjusted R- squared: F- sta:s:c: on 1 and 1221 DF, p- value: < 2.2e- 16 Important numbers: b 0 = b 1 = Fi@ed line: y= x SE(b 1 )= n- 2=1221 R 2 =16.63% 1. What is the correla:on between birth weight and length of gesta:on? 2. What is ? 3. What is a 95% CI for the slope? (You can ignore: Adjusted R- squared F- sta:s:c line)

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4 line: y= x What is the es:mated birth weight for a baby with a gesta:on period of 300 days? (300)=128 ounces The slope of a least squares regression line is interpreted as the predicted change in the response variable associated with a one unit increase in the explanatory variable. For each increase of 1 day in gesta:on period, birth weight increases by 0.46 ounces, on average. For each 10 day increase in gesta:on period, birth weight increases by 4.6 ounces, on average. The intercept of a least squares regression line is interpreted as the predicted value of the response variable when the explanatory variable has a value of zero. In many context this predic:on makes no sense. This example is one of those cases when this predic:on makes no sense. In most situa:ons, it is dangerous to use the regression line to make predic:ons outside the range of the observed values of the explanatory variable. This is called extrapola4on. You can t be sure that the same linear rela:onship holds outside the range of your observed data so you shouldn t trust extrapola:ons.

5 Is a linear model a good fit for this rela4onship? Assump:ons of linear regression, necessary to carry out inference on the slope: 1. A linear model is appropriate for the data. 2. The observa:ons are independent of each other. 3. The variability in the residuals is the same for all values of the explanatory variable. 4. The residuals are normally distributed. How to check these assump:ons: 1. Look at a sca@erplot of the data. Look at a sca@erplot of the residuals versus the explanatory variable or versus the fi@ed values. 2. Understand how the data were collected. 3. Look at a sca@erplot of the residuals versus the explanatory variable or versus the fi@ed values. 4. Look at a normal quan:le plot of the residuals. A residual plot is a sca@erplot of the residuals versus either the explanatory variable or the fi@ed values. If a straight line is an appropriate model, this should look like random sca@er. Poten:al problems that can be seen in a residual plot include curvature, non- constant variability, and outliers. A normal quan4le plot is a plot of the residuals versus the corresponding quan:les from a normal distribu:on. If the residuals are normally distributed, it looks like a straight line.

6 plot with the regression line Are the two points with small gesta1on 1mes affec1ng the suitability of a linear model for these data? birth weight (in ounces) length of gestation period (days)

7 Plot of residuals versus explanatory variable Plot of residuals versus values residuals residuals length of gestation period (days) fitted values No curvature Variability appears to be approximately constant Random sca@er except for the two points with small gesta:on periods

8 Normal quantile plot of residuals Sample Quantiles Looks good! Theoretical Quantiles

9 An observa:on is influen4al if removing it from the dataset substan:ally changes the slope and/or intercept of the least squares regression line. Typically, points that are outliers in the explanatory variable have the poten:al to be influen:al. birth weight (in ounces) Are the points with small gesta1on periods influen1al? length of gestation period (days)

10 Re- do the regression with the two points with small gesta:on periods removed. R output: Call: lm(formula = bwt_2_pts_removed ~ gest_2_pts_removed) Residuals: Min 1Q Median 3Q Max Coefficients: Es:mate Std. Error t value Pr(> t ) (Intercept) * gest_2_pts_removed <2e- 16 *** Signif. codes: 0 *** ** 0.01 * birth weight (in ounces) length of gestation period (days) Residual standard error: on 1219 degrees of freedom (13 observa:ons deleted due to missingness) Mul:ple R- squared: , Adjusted R- squared: F- sta:s:c: on 1 and 1219 DF, p- value: < 2.2e- 16 Previous line: y= x (es:mate of the intercept has changed by 10!)

11 More on Normal quan1le plots: Match the quan:le plot to the histogram Frequency Frequency Frequency Frequency heavytail x normal100 outlier Normal Q-Q Plot Normal Q-Q Plot Normal Q-Q Plot Normal Q-Q Plot Sample Quantiles Sample Quantiles Sample Quantiles Sample Quantiles Theoretical Quantiles Theoretical Quantiles Theoretical Quantiles Theoretical Quantiles A B C D ANSWER: 1- D (heavy tails), 2- A (right- skewed), 3- B (normal), 4- C (outlier)

12 Always be careful with your conclusions from regression Some reasons two quan:ta:ve variables may be related: The explanatory variable directly causes the response variable. The response variable causes a change in the explanatory variable. The explanatory variable contributes to, but is not the sole cause of the response variable. Confounding variables exist. Both variables result from a common cause. Both variables are changing over :me. The rela:onship is just a coincidence.

13 Another rela1onship to inves1gate from the Child Health and Development Study: Is there a rela:onship between the mother s age and the baby s birth weight? birth weight (in ounces) mother's age (in years)

14 For the rela:onship between mother s age and baby s birth weight, what might you say about: R 2 close to 0 The slope close to 0 birth weight (in ounces) mother's age (in years) The P- value t- test for the two- sided test that the slope is 0 large A confidence interval for the slope includes 0 The plot of the residuals versus the fi@ed values looks like random sca@er (no indica:on of a non- linear rela:onship nor non- constant variability) The normal quan:le plot of the residuals Hard to say. No reason to expect that they wouldn t be normally distributed (and then the plot would be a straight line).

15 An example of how regression can be misused and R 2 isn t the en4re story: Old Faithful, geyser in Yellowstone Na:onal Park, California Visitors come to see it erupt, so park rangers want to be able to predict when the next erup:on will take place. Photos from: h@p://

16 An example of how regression can be misused and R 2 isn t the en4re story: Old Faithful, geyser in Yellowstone Na:onal Park, California Visitors come to see it erupt, so park rangers want to be able to predict when the next erup:on will take place. interval duration

17 An example of how regression can be misused and R 2 isn t the en4re story: Old Faithful, geyser in Yellowstone Na:onal Park, California Visitors come to see it erupt, so park rangers want to be able to predict when the next erup:on will take place. interval R 2 =85% duration Is the regression line giving good predic1ons?

18 An example of how regression can be misused and R 2 isn t the en4re story: Old Faithful, geyser in Yellowstone Na:onal Park, California Visitors come to see it erupt, so park rangers want to be able to predict when the next erup:on will take place. interval R 2 =2% R 2 =7% duration

19 An example of how regression can be misused and R 2 isn t the en4re story: Old Faithful, geyser in Yellowstone Na:onal Park, California Visitors come to see it erupt, so park rangers want to be able to predict when the next erup:on will take place. Rather than using the regression R 2 model, park rangers found that they =7% could predict the :me of the next erup:on just as well by saying it will be 90 minutes if the previous erup:on was long and 60 minutes if the previous erup:on was short in dura:on. interval R 2 =2% duration Moral of the story: Look at the data first. R 2 is not the whole story. Two separated groups of points, each with no rela:onship between the response and explanatory variable, can result in a regression with high R 2 when the two groups are considered together.