2 In Chapter 14: 14.1 Data 14.2 Scatterplots 14.3 Correlation 14.4 Regression
3 14.1 Data Quantitative response variable Y (“dependent variable”) Quantitative explanatory variable X (“independent variable”) Historically important public health data set used to illustrate techniques (Doll, 1955) –n = 11 countries –Explanatory variable = per capita cigarette consumption in 1930 (CIG1930) –Response variable = lung cancer mortality per 100,000 (LUNGCA)
5 §14.2 Scatterplot Bivariate (x i, y i ) points plotted as scatter plot.
6 Inspect scatterplot’s Form: Can the relation be described with a straight or some other type of line? Direction: Do points tend trend upward or downward? Strength of association: Do point adhere closely to an imaginary trend line? Outliers (in any): Are there any striking deviations from the overall pattern?
7 Judging Correlational Strength Correlational strength refers to the degree to which points adhere to a trend line The eye is not a good judge of strength. The top plot appears to show a weaker correlation than the bottom plot. However, these are plots of the same data sets. (The perception of a difference is an artifact of axes scaling.)
8 §14.3. Correlation Correlation coefficient r quantifies linear relationship with a number between −1 and 1. When all points fall on a line with an upward slope, r = 1. When all data points fall on a line with a downward slope, r = −1 When data points trend upward, r is positive; when data points trend downward, r is negative. The closer r is to 1 or −1, the stronger the correlation.
10 Calculating r Formula Correlation coefficient tracks the degree to which X and Y “go together.” Recall that z scores quantify the amount a value lies above or below its mean in standard deviations units. When z scores for X and Y track in the same direction, their products are positive and r is positive (and vice versa).
12 Calculating r In practice, we rely on computers and calculators to calculate r. I encourage my students to use these tools whenever possible.
13 Calculating r SPSS output for Analyze > Correlate > Bivariate using the illustrative data:
14 Interpretation of r 1.Direction. The sign of r indicates the direction of the association: positive (r > 0), negative (r < 0), or no association (r ≈ 0). 2.Strength. The closer r is to 1 or −1, the stronger the association. 3.Coefficient of determination. The square of the correlation coefficient (r 2 ) is called the coefficient of determination. This statistic quantifies the proportion of the variance in Y [mathematically] “explained” by X. For the illustrative data, r = 0.737 and r 2 = 0.54. Therefore, 54% of the variance in Y is explained by X.
15 Notes, cont. 4. Reversible relationship. With correlation, it does not matter whether variable X or Y is specified as the explanatory variable; calculations come out the same either way. [This will not be true for regression.] 5. Outliers. Outliers can have a profound effect on r. This figure has an r of 0.82 that is fully accounted for by the single outlier.
16 Notes, cont. 6. Linear relations only. Correlation applies only to linear relationships This figure shows a strong non-linear relationship, yet r = 0.00. 7. Correlation does not necessarily mean causation. Beware lurking variables (next slide).
17 Confounded Correlation A near perfect negative correlation (r = −.987) was seen between cholera mortality and elevation above sea level during a 19th century epidemic. We now know that cholera is transmitted by water. The observed relationship between cholera and elevation was confounded by the lurking variable proximity to polluted water.
18 Hypothesis Test We conduct the hypothesis test to guard against identifying too many random correlations. Random selection from a random scatter can result in an apparent correlation
19 Hypothesis Test A.Hypotheses. Let ρ represent the population correlation coefficient. H 0 : ρ = 0 vs. H a : ρ ≠ 0 (two-sided) [or H a : ρ > 0 (right-sided) or H a : ρ < 0 (left-sided)] B.Test statistic C.P-value. Convert t stat to P-value with software or Table C.
20 Hypothesis Test – Illustrative Example A.H 0 : ρ = 0 vs. H a : ρ ≠ 0 (two-sided) B.Test stat C..005 < P <.01 by Table C. P =.0097 by computer. The evidence against H 0 is highly significant.
23 Conditions for Inference Independent observations Bivariate Normality (r can still be used descriptively when data are not bivariate Normal)
24 §14.4. Regression Regression describes the relationship in the data with a line that predicts the average change in Y per unit X. The best fitting line is found by minimizing the sum of squared residuals, as shown in this figure.
25 Regression Line, cont. The regression line equation is: where ŷ ≡ predicted value of Y, a ≡ the intercept of the line, and b ≡ the slope of the line Equations to calculate a and b SLOPE: INTERCEPT:
26 Regression Line, cont. Slope b is the key statistic produced by the regression
27 Regression Line, illustrative example Here’s the output from SPSS:
28 Let α represent the population intercept, β represent population slope, and ε i represent the residual “error” for point i. The population regression model is The estimated standard error of the regression is A (1−α)100% CI for population slope β is Inference
32 Analysis of Variance of the Regression Model An ANOVA technique equivalent to the t test can also be used to test H 0 : β = 0. This technique is covered on pp. 321 – 324 in the text but is not included in this presentation.
33 Conditions for Inference Inference about the regression line requires these conditions Linearity Independent observations Normality at each level of X Equal variance at each level of X
34 Conditions for Inference This figure illustrates Normal and equal variation around the regression line at all levels of X
35 Assessing Conditions The scatterplot should be visually inspected for linearity, Normality, and equal variance Plotting the residuals from the model can be helpful in this regard. The table lists residuals for the illustrative data
36 Assessing Conditions, cont. A stemplot of the residuals show no major departures from Normality This residual plot shows more variability at higher X values (but the data is very sparse) |-1|6 |-0|2336 | 0|01366 | 1|4 x10
37 Residual Plots With a little experience, you can get good at reading residual plots. Here’s an example of linearity with equal variance.
38 Residual Plots Example of linearity with unequal variance
39 Example of Residual Plots Example of non-linearity with equal variance