BIOS 501 Lecture 2 Plotting relationships Roderick Little

BIOS 501 Lecture 2 Plotting relationships Roderick Little UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Looking at Data-Relationships o o Scatterplots of two variables to visualize relationships Measuring degree of association: Correlation Response vs explanatory variables Fitting a regression line n See IPS Sections 2. 1 -2. 5 2 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Scatterplots Example 1: Mean height of a group of children in Kalama, Egypt surveyed when 18 to 29 months old. Plot the (x, y) pairs: Y vs. X Y=? X=? 3 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Example 1: Mean height of a group of children in Kalama, Egypt, plotted against age from 18 to 29 months. 4 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Assessing relationships o o Direction: does Y increase with X (positive association), Y decrease with X (negative association)? What is the shape of the underlying form (linear, parabolic, exponential, etc. )? Strength: do the points follow the underlying form quite closely or is there large scatter? Outliers: isolated deviations from overall relationship? 5 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Example 1: Mean height of a group of children in Kalama, Egypt, plotted against age from 18 to 29 months. Positive association Linear, strong Scatter about line is small 6 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Example 2: State mean SAT math score plotted against percent of HS seniors taking the exam 7 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Negative relationship Non-linear Moderate scatter 8 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Association is not causality o o Caution: often spurious, other variables lurking in the background Shorter women have lower risk of heart attack Countries with more TV sets have better life expectancy rates Can’t tell whether a relationship is causal from the plot – depends on other considerations 9 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

More on scatter plots o Plotting different categories with different symbols may throw light on data 10 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Example 2: Adding region via plotting symbol: e is for northeastern states and m is for Midwestern states (others excluded). May enhance understanding of the data. 11 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

More on scatter plots o Plotting on a restricted scale magnifies the relationship, but potential to mislead. 12 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Effect of changing scales 13 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

2. 2 Correlation Coefficient o o Linear relationships are quite common. Correlation coefficient r measures strength and direction of a linear relationship between two quantitative variables X and Y. Data structure: (X, Y) pairs measured on n individuals (Weight, blood pressure) or (age, bone-density) measured on a set of subjects 14 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

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Correlation o o o Lies between -1 and 1 You can switch roles of X and Y, r will remain the same. Unit free, unaffected by linear transformation r positive for positive association, negative for negative association X and Y should both be quantitative. r near 0 implies weak linear relationship, closer to +1 or -1 suggests very strong linear pattern. 16 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

More on correlation o o r is sensitive to outliers. Captures only the strength of “linear” relationship, it could be true that Y and X have a very strong curvilinear relationship but r is close to zero. r = +1 or -1 only when points lie perfectly on a straight line (e. g. Y =2 X+3). In SPSS->Analyze->Correlation->Bivariate 17 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Formula: 18 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

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A strong but highly nonlinear relationship: low correlation 20 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

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Summary o o o Scatterplots: look form, direction, strength, outliers Correlation: numerical measure capturing direction and strength of relationship Sign of r: direction Value of r: strength Always: Plot the data, look at other descriptive measures along with the correlation. 23 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

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2. 3 Regression line o Straight line which describes best how the response variable Y changes when the explanatory variable X changes n o Equation of straight lines: n o (We do distinguish between Y and X, cannot switch their roles) y = a+bx (b is the slope, a is the intercept) Task: calculate best a and b for your data. n n Find the line that best fits your data. Use this line to predict y for different values of x. 26 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Example 1: Regression line for Kamala, Egypt, and a prediction based on the line (mean height at 32 months). Positively associated. Positive slope b. 27 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Prediction and Extrapolation o Fitted line for Kalama data: n o Prediction at 32 months: n o 64. 93+. 635* 32=85. 25 Prediction at 20 years (240 months) n o Height=64. 93 +. 635 * Age in months 64. 93+. 635*240=217. 33 cm = about 7 feet Extrapolation beyond the range of data: dangerous! 28 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Least squares regression line o o The line that makes the sum of squares of the vertical distances of the data points from the line as small as possible Observed response = y, predicted response by using the line = yhat Error in prediction: residual = y – yhat Least squares line minimizes sum of squared residuals – see text for detailed formulae. 29 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Example 3: Regression line gesell score against age when first word was pronounced. 30 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

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Properties of regression line o The slope b measures the change in the mean of Y for a one unit increase in X. n n o Slope depends on which variable is X and which is Y. n o E. g. , for Egypt example, height changes an average of. 635 cm for every 1 month increase in age. How about 5 month increase in age? Regression lines of Y on X and X on Y differ. Line passes through the mean of X and mean of Y. 33 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Slope and correlation o o Sign of b is same as the sign of r, indicates the direction of linear association. b is unchanged by adding or subtracting a constant from values of X and Y, but is affected by scale changes in X and Y: n n n E. g. convert: X = age from months to years Slope when X = age in months = 2 (say) Slope when X = age in years= ? 34 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Correlation and Regression o o o The square of correlation, r 2, is the fraction of variation in Y explained by the least squares regression of Y on X. Higher r 2 indicates better fit. r 2=. 989 for Egypt data means 98. 9% of the unexplained variation in height is explained by the fitted regression line with X = age. 35 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

r 2 measures how much of the variation in Y=height is explained by the linear relationship between Y=height and X=age a Picture (a): less spread, tight fit, r 2=0. 989 b Picture (b): more scatter, more error in prediction, r 2=0. 849 36 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Association is not necessarily causation o o Common problem: Two variables are related because a third one influences them (lurking variable). Confounding: Effects of two variables cannot be distinguished. 37 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Examples o 1. Does birth control reduce fertility in developing countries? n n n o X = use of birth control Y = number of children Expect to see negative association, but in fact association is positive! Confounding variable is number of children at time of birth control – women with many children more likely to use birth control 2. Do higher nursing staff levels reduce “failure to rescue” events in hospitals? n n X = nursing staff level Y = incidence of “failure to rescue” events Confounding variable: Z = severity of illness 38 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

3. Does going to church increase life expectancy? n n n x = attending religious services y = how long the person lives z = general good habits and discipline 39 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

4. Does marriage cure poverty? o o “The truth is, the greatest tool to lift children and families from poverty is one that decreases the probability of child poverty by 82%. But it isn’t a government spending program. It’s called marriage. ” U. S. Senator Marco Rubio “It isn’t that having a lasting and successful marriage is a cure for living in poverty… Living in poverty is a barrier to having a lasting and successful marriage. ” Kristi Williams, Ohio State University , quoted in “For Richer, for Poorer” by Annie Lowrey, NYT magazine Feb 9, 2014. 40 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH

Summary o o Use scatterplots, correlation and regression to capture association between two quantitative variables. Distinguish between association and causation. n o There could be more than one explanation of association. More on regression later. 41 UNIVERSITY OF MICHIGAN SCHOOL OF PUBLIC HEALTH