Chapters 2 and 10: Least Squares Regression
Learning goals for this chapter:
Describe the form, direction, and strength of a scatterplot. Use SPSS output to find the following: least-squares regression line, correlation, r2, and estimate for σ. Interpret a scatterplot, residual plot, and Normal probability plot. Calculate the predicted response and residual for a particular x-value. Understand that least-squares regression is only appropriate if there is a linear relationship between x and y. Determine explanatory and response variables from a story. Use SPSS to calculate a prediction interval for a future observation. Perform a hypothesis test for the regression slope and for zero population correlation/independence, including: stating the null and alternative hypotheses, obtaining the test statistic and P-value from SPSS, and stating the conclusions in terms of the story. Understand that correlation and causation are not the same thing. Estimate correlation for a scatterplot display of data. Distinguish between prediction and extrapolation. Check for differences between outliers and influential outliers by rerunning the regression. Know that scatterplots and regression lines are based on sample data, but hypothesis tests and confidence intervals give you information about the population parameter.
When you have 2 quantitative variables and you want to look at the relationship between them, use a scatterplot. If the scatter plot looks linear, then you can do least squares regression to get an equation of a line that uses x to explain what happens with y.
The general procedure:
1. Make a scatter plot of the data from the x and y variables. Describe the form, direction, and strength. Look for outliers.
2. Look at the correlation to get a numerical value for the direction and strength.
3. If the data is reasonably linear, get an equation of the line using least squares regression.
4. Look at the residual plot to see if there are any outliers or the possibility of lurking variables. (Patterns bad, randomness good.)
1 5. Look at the normal probability plot to determine whether the residuals are normally distributed. (The dots sticking close to the 45-degree line is good.)
6. Look at hypothesis tests for the correlation, slope, and intercept. Look at confidence intervals for the slope, intercept, and mean response, and at the prediction intervals.
7. If you had an outlier, you should re-work the data without the outlier and comment on the differences in your results.
Association
Positive, negative, or no association Remember: ASSOCIATON or CORRELATION is NOT the same thing as CAUSATION. (See chapter 3/2.5 notes.)
Response variable: Y Dependent variable measures an outcome of a study
Explanatory variable: X Independent variable explains or is related to changes in the response variables (p. 105)
Scatterplots: Show the relationship between 2 quantitative variables measured on the same individuals
Dots only—don’t connect them with a line or a curve
Form: Linear? Non-linear? No obvious pattern?
Direction: Positive or negative association? No association?
Strength: how closely do the points follow a clear form? Strong or weak or moderate?
Look for OUTLIERS!
Correlation: measures the direction and strength of the linear relationship between 2 quantitative variables, r. It is the standardized value for each observation with respect to the mean and standard deviation.
2 1 x x y y r ii where we have data on variables x and y for n individuals. n1 sxy s You won’t need to use this formula, but SPSS will.
Using SPSS to get correlation: Use the Pearson Correlation output. Analyze --> Correlate --> Bivariate (see page 55 in the SPSS manual). The SPSS manual tells you where to find r using the least squares regression output, but this r is actually the ABSOLUTE VALUE OF r, so you need to pay attention to the direction yourself. The Pearson Correlation gives you the actual r with the correct sign.
Properties of correlation:
X and Y both have to be quantitative. It makes no difference which you call X and which you call Y. Does not change when you change the units of measurement. If r is positive, there is a positive association between X and Y As X increases, Y increases If r is negative, there is a negative association between X and Y As X increases, Y decreases 11r The closer r is to –1 or to 1, the stronger the linear relationship The closer r is to 0, the weaker the linear relationship Outliers strongly affect r. Use r with caution if outliers are present.
3 Example: We want to examine whether the amount of rainfall per year increases or decreases corn bushel output. A sample of 10 observations was taken, and the amount of rainfall (in inches) was measured, as was the subsequent growth of corn.
Amount of Rain Bushels of Corn 3.03 80 3.47 84 4.21 90 4.44 95 4.95 97 5.11 102 5.63 105 6.34 112 6.56 115 6.82 115
The scatterplot:
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corn yield (bushels) 70 2 3 4 5 6 7
amount of rain (in) a) What does the scatterplot tell us? What is the form? Direction? Strength? What do we expect the correlation to be?
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Correlations
amount of corn yield rain (in) (bushels) amount of rain (in) Pearson Correlation 1 .995(**) Sig. (2-tailed) . .000 N 10 10 corn yield (bushels) Pearson Correlation .995(**) 1 Sig. (2-tailed) .000 . N 10 10 ** Correlation is significant at the 0.01 level (2-tailed).
Inference for Correlation:
R = correlation
R2 = % of variation in Y explained by the regression line (the closer to 100%, the better)
ρ (Greek letter rho) = correlation for the population
When ρ = 0, there is no linear association in the population, so X and Y are independent (if X and Y are both normally distributed).
Hypothesis test for correlation:
rn 2 To test the null hypothesis H0: ρ = 0, SPSS will compute the t statistic: t , 1 r 2 degrees of freedom = n – 2 for simple linear regression.
b) Are corn yield and rain independent in the population? Perform a test of significance to determine this.
c) Do corn yield and rain have a positive correlation in the population? Perform a test of significance to determine this.
This test statistic for the correlation is numerically identical to the t statistic used to test H0: 1 = 0.
Can we do better than just a scatter plot and the correlation in describing how x and y are related? What if we want to predict y for other values of x?
5 Least-Squares Regression fits a straight line through the data points that will minimize the sum of the vertical distances of the data points from the line. n 2 Minimizes ()ei i 1
Equation of the line is: yˆˆ b01 b x, with y = the predicted yline
sy Slope of the line is: br1 , where the slope measures the amount of change sx caused in the predicted response variable when the explanatory variable is increased by one unit.
Intercept of the line is: b01 y b x , where the intercept is the value of the predicted response variable when the explanatory variable = 0.
Type of line Least Squares Regression slope y-intercept equation of line
Ch. 10 Sample yˆ b01 b x b1 b0
Ch. 10 Population yxi01 i i 1 0 (model)
Using the corn example, find the least squares regression line. Tell SPSS to do AnalyzeRegression Linear. Put “rain” into the independent box and “corn” into the dependent box. Click OK. Model Summaryb
Adjusted Std. Error of Model R R Square R Square the Estimate 1 .995a .991 .989 1.290 a. Predictors: (Constant), amount of rain (in) b. Dependent Variable: corn yield (bushels) ANOVAb
Sum of Model Squares df Mean Square F Sig. 1 Regression 1397.195 1 1397.195 840.070 .000a Residual 13.305 8 1.663 Total 1410.500 9 a. Predictors: (Constant), amount of rain (in) b. Dependent Variable: corn yield (bushels) Coe fficientsa
Unstandardized Standardized Coefficients Coefficients 95% Confidence Interval for B Model B Std. Error Beta t Sig. Lower Bound Upper Bound 1 (Constant) 50.835 1.728 29.421 .000 46.851 54.819 amount of rain (in) 9.625 .332 .995 28.984 .000 8.859 10.391 a. Dependent Variable: corn yield (bushels)
6 d) What is the least-squares regression line equation?
The scatterplot with the least squares regression line looks like:
120 R2 is the percent of
110 variation in corn yield explained by the regression line with rain= 99.06% 100
90
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corn yield (bushels) 70 Rsq = 0.9906 2 3 4 5 6 7
amount of rain (in)
Hypothesis testing for H0: 1 = 0
b Test statistic: t 1 with df = n - 2 SEb1
SPSS will give you the test statistic (under t), and the 2-sided P-value (under Sig.).
e) Is the slope positive in the population? Perform a test of significance.
f) What % of the variability in corn yield is explained by the least squares regression line?
g) What is the estimate of the standard error of the model?
7 What do we mean by prediction or extrapolation? Use your least-squares regression line to find y for other x-values. Prediction: using the line to find y-values corresponding to x-values that are within the range of your data x-values. Extrapolation: using the line to find y-values corresponding to x-values that are outside the range of your data x-values. Be careful about extrapolating y-values for x-values that are far away from the x data you currently have. The line may not be valid for wide ranges of x!
Example: On the rain/corn data above, predict the corn yield for
a) 5 inches of rain
b) 7.2 inches of rain
c) 0 inches of rain
d) 100 inches of rain
e) For which amounts of rainfall above do you think the line does a good job of predicting actual corn yield? Why?
Cartoon by J.B. Landers on www.causeweb.org (used with permission)
8 Prediction Intervals Predicting a future observation under conditions similar to those used in the study. Since there is variability involved in using a model created from sample data, a prediction interval is better than a single prediction. They’re related to confidence intervals. Use SPSS.
The 95% prediction interval for future corn yield measurements when rain = 5.11 is (96.90, 103.14).
Assumptions for Regression:
1. Repeated responses y are independent of each other. 2. For any fixed value of x, the response y varies according to a Normal distribution. 3. The mean response has a straight-line relationship with x. y 4. The standard deviation of y (σ) is the same for all values of x. The value of σ is unknown.
How do you check these assumptions?
Scatterplot and R2: Do you have a straight-line relationship between X and Y? How strong is it? How close to 100% is R2? Hopefully no outliers! (#3)
9 Normal probability plot: Are the residuals approximately normally distributed? Do the dots fall fairly close to the diagonal line (which is always there in the same spot)? (#2) Normal P-P Plot of Regression Standardized Residual
Dependent Variable: corn yield (bushels) 1.0
0.8
0.6
0.4 Expected Cum Prob Cum Expected
0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0 Observed Cum Prob Residual plot: Do you have constant variability? Do the dots on your residual plot look random and fairly evenly distributed above and below the 0 line? Hopefully no outliers! (#1 and 4)
Residual is the vertical difference between the observed y-value and the regression line y-value:
residual ei y i yˆ i y i a bx i y data y line
Residual plot: scatterplot of the regression residuals against the explanatory variable (e vs. x) e-axis has both negative and positive values but centered about e = 0. the mean of the least-squares residuals is always zero. e 0 Good: total randomness, no pattern, approximately the same number of points above and below the e = 0 line Bad: obvious pattern, funnel shape, parabola, more points above 0 than below (or vice versa) if you have a pattern, your data does not necessarily fit the model (line) well
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Example: Show a residual plot for the corn/rain data using SPSS.
2.5
2.0
1.5
1.0
.5
0.0
-.5
-1.0
-1.5
Unstandardized Residual -2.0 2 3 4 5 6 7
amount of rain (in)
Outliers: Outliers are observations that lie outside the overall pattern of the other observations. Outliers in the y direction of a scatterplot have large regression residuals (ei) Outliers in the x direction of a scatterplot are often influential for the regression line An observation is influential if removing it would markedly change the result of the calculation Outliers can drastically affect regression line, correlation, means, and standard deviations. You can draw a second regression line that doesn’t include the outliers—if the second line moves more than a small amount when the point is deleted or if R2 changes much, the point is influential
Which hypothesis test do you use when? If you’re not sure whether to use β1 or ρ, here are some guidelines. The test statistics and P-values are identical for either symbol.
Use If the words are: β1 Slope, regression coefficient ρ Correlation, independence Either β1 or ρ linear relationship
11 Review of SPSS instructions for Regression: When you set up your regression, you click on:
Analyze-->Regression-->Linear. Put in your y variable for "dependent" and your x variable for "independent" on the gray screen. Don't hit "ok" yet though.
Back on the regression gray screen, click on "Plots", and then click on "normal probability plot." Click "continue" on the Plots gray screen.
Back on the regression gray screen, click on "Save", and then click on “unstandardized residuals." Click “Individual” under the “Prediction Interval” section, and adjust the confidence level, if needed. Click "continue" on the Save gray screen and then "ok" to the big Regression gray screen.
The prediction interval and the residuals will show up back on the data input screen. The LICI_1 and UICI_1 give you the prediction interval lower and upper bounds.
You still won't have a residual plot yet. If you click back to your data input screen, you now have a new column called "Res_1". To make the residual plot, you follow the same steps for making a scatterplot: go to graphs-->scatter-->simple, then put "Res_1" in for y and your x variable in for x. Click "ok." Once you see your residual plot, you'll need to double click on it to go to Chart Editor. On the Chart Editor tool bar, you can see a button that shows a graph with a horizontal line. Click on that button. Make sure that the y- axis is set to 0.
12 Example: The scatterplot below shows the calories and sodium content for each of 17 brands of meat hot dogs.
a) Describe the main features of the relationship.
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Sodium content 100 100 120 140 160 180 200
Calories
b) What is the correlation between calories and sodium?
Correlations
Sodium Calories content Calories Pearson Correlation 1 .863** Sig. (2-tailed) . .000 N 17 17 Sodium content Pearson Correlation .863** 1 Sig. (2-tailed) .000 . N 17 17 **. Correlation is significant at the 0.01 level (2-tailed).
c) Report the least-squares regression line.
Model Summaryb
Adjusted Std. Error of Model R R Square R Square the Estimate 1 .863a .745 .728 48.913 a. Predictors: (Constant), Calories b. Dependent Variable: Sodium content
14 Coe fficientsa
Unstandardized Standardized Coefficients Coefficients 95% Confidence Interval for B Model B Std. Error Beta t Sig. Lower Bound Upper Bound 1 (Constant) -91.185 77.812 -1.172 .260 -257.038 74.668 Calories 3.212 .485 .863 6.628 .000 2.179 4.245 a. Dependent Variable: Sodium content
d) Show a residual plot and comment on its features.
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Unstandardized Residual -200 100 120 140 160 180 200
Calories
e) Is there an outlier? If so, where is it?
f) Show a normal probability plot and comment on its features.
Normal P-P Plot of Regression Standardized Residual
Dependent Variable: Sodium content 1.0
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0.6
0.4
Expected Cum Prob Cum Expected
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0.0 0.0 0.2 0.4 0.6 0.8 1.0 Observed Cum Prob 15
g) Leave off the outlier, and recalculate the correlation and another least- squares regression line. Is your outlier influential? Explain your answer.
Correlations
cal2 sod2 cal2 Pearson Correlation 1 .834** Sig. (2-tailed) . .000 N 16 16 sod2 Pearson Correlation .834** 1 Sig. (2-tailed) .000 . N 16 16 **. Correlation is significant at the 0.01 level (2-tailed).
Model Summaryb
Adjusted Std. Error of Model R R Square R Square the Estimate 1 .834a .695 .674 36.406 a. Predictors: (Constant), cal2 b. Dependent Variable: sod2
Coe fficientsa
Unstandardized Standardized Coefficients Coefficients 95% Confidence Interval for B Model B Std. Error Beta t Sig. Lower Bound Upper Bound 1 (Constant) 46.900 69.371 .676 .510 -101.886 195.686 cal2 2.401 .425 .834 5.653 .000 1.490 3.312 a. Dependent Variable: sod2
h) If there is a new brand of meat hot dog with 150 calories per frank, how many milligrams of sodium do you estimate that one of these hotdogs contains?
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