Presentation on theme: "Two Quantitative Variables"— Presentation transcript:
1Two Quantitative Variables Chapter 10Two Quantitative VariablesInference for Correlation: Simulation-Based MethodsLeast-Squares RegressionInference for Correlation and Regression: Theory-Based Methods
2Section 10.1 and 10.2: Summarizing Two Quantitative Variables and Inference for Correlation (Simulation)I was interested in knowing the relationship between the time it takes a student to take a test and the resulting score. So I collected some data …Time30414347485154565758Score100849490889985656489606162636669727879838692747375539168
3Scatterplot Put explanatory variable on the horizontal axis. Put response variable on the vertical axis.
4Describing Scatterplots When we describe data in a scatterplot, we describe theDirection (positive or negative)Form (linear or not)Strength (strong-moderate-weak, we will let correlation help us decide)How would you describe the time and test scatterplot?
5CorrelationCorrelation measures the strength and direction of a linear association between two quantitative variables.Correlation is a number between -1 and 1.With positive correlation one variable increases, on average, as the other increases.With negative correlation one variable decreases, on average, as the other increases.The closer it is to either -1 or 1 the closer the points fit to a line.The correlation for the test data is
6Correlation Guidelines Correlation ValueStrength of AssociationWhat this means0.5 to 1.0StrongThe points will appear to be nearly a straight line0.3 to 0.5ModerateWhen looking at the graph the increasing/decreasing pattern will be clear, but it won’t be nearly a line0.1 to 0.3WeakWith some effort you will be able to see a slightly increasing/decreasing pattern0 to 0.1NoneNo discernible increasing/decreasing patternSame Strength Results with Negative Correlations
7Back to the test dataI was not completely honest. There were three more test scores. The last three people to finish the test had scores of 93, 93, and 97.What does this do to the correlation?
8Influential Observations The correlation changed from (a fairly strong negative correlation) to (a fairly weak negative correlation).Points that are far to the left or right and not in the overall direction of the scatterplot can greatly change the correlation. (influential observations)
9CorrelationCorrelation measures the strength and direction of a linear association between two quantitative variables.-1 < r < 1Correlation makes no distinction between explanatory and response variables.Correlation has no unit.Correlation is not a resistant measure.Correlation based on averages is usually higher than that for individuals.Correlation game
10Inference for Correlation with Simulation 1. Compute the observed statistic. (Correlation)2. Scramble the response variable, compute the simulated statistic, and repeat this process many times.3. Reject the null hypothesis if the observed statistic is in the tail of the null distribution.
11Temperature and Heart Rate HypothesesNull: There is no association between heart rate and body temperature. (ρ = 0)Alternative: There is a positive linear association between heart rate and body temperature. (ρ > 0)ρ = rho
12Temperature and Heart Rate Collect the DataTmp98.398.298.798.597.098.899.397.8HR726971808168826599.998.698.497.496.798.07986588473576289
13Temperature and Heart Rate Explore the Datar = 0.378
14Temperature and Heart Rate If there was no association between heart rate and body temperature, what is the probability we would get a correlation as high as just by chance?If there is no association, we can break apart the temperatures and their corresponding heart rates. We will do this by shuffling one of the variables. (The applet shuffles the response.)
15Shuffled r =-0.216Shuffled r =0.212Shuffled r =-0.097Temp HR97 7998 68Temp HR97 7198 65Temp HR97 7398 72
16Temperature and Heart Rate In three shuffles, we obtained correlations of , and These are all a bit closer to 0 than our sample correlation of 0.378Let’s scramble 1000 times and compute 1000 simulated correlations.
17Temperature and Heart Rate Notice our null distribution is centered at 0 and somewhat symmetric.We found that 530/10000 times we had a simulated correlation greater than or equal to
18Temperature and Heart Rate With a p-value of we don’t have strong evidence there is a positive linear association between body temperature and heart rate. However, we do have moderate evidence of such an association and perhaps a larger sample give a smaller p-value.If this was significant, to what population can we make our inference?
19Temperature and Heart Rate Let’s look at a different data set comparing temperature and heart rate (Example 10.5A) using the Analyzing Two Quantitative Variables applet.Pay attention to which variable is explanatory and which is response.The default statistic used is not correlation, so we need to change that.
20Work on exploration 10.2 to see if there is an association between birth date and the 1970 draft lottery number.
22IntroductionIf we decide an association is linear, it’s helpful to develop a mathematical model of that association.Helps make predictions about the response variable.The least-squares regression line is the most common way of doing this.
23IntroductionUnless the points form a perfect line, there won’t be a single line that goes through every point.We want a line that gets as close as possible to all the points
24IntroductionWe want a line that minimizes the vertical distances between the line and the pointsThese distances are called residuals.The line we will find actually minimizes the sum of the squares of the residuals.This is called a least-squares regression line.
26Growing Plates?There are many recent articles and TV reports about the obesity problem.One reason some have given is that the size of dinner plates are increasing.Are these black circles the same size, or is one larger than the other?
27Growing Plates?They appear to be the same size for many, but the one on the right is about 20% larger than the left.This suggests that people will put more food on larger dinner plates without knowing it.
28Portion DistortionThere is name for this phenomenon: Delboeuf illusion
29Growing Plates?Researchers gathered data to investigate the claim that dinner plates are growingAmerican dinner plates sold on ebay on March 30, 2010 (Van Ittersum and Wansink, 2011)Size and year manufactured.Year1950195619571958196319641969197419751978size1010.810.110.610.519801986199019952004200720082009Size10.3810.41111.511.1
30Growing Plates?Both year (explanatory variable) and size (response variable) are quantitative.Each dot represents one plate in this scatterplot.Describe theassociationhere.
31Growing Plates? The association appears to be roughly linear The least squares regression line is addedHow can we describe this line?
32Growing Plates? The regression equation is 𝑦 =𝑎+𝑏𝑥: a is the y-interceptb is the slopex is a value of the explanatory variableŷ is the predicted value for the response variableFor a specific value of x, the corresponding distance y− 𝑦 is a residual
33Growing Plates?The least squares line for the dinner plate data is 𝑦 =− 𝑥Or 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 =− (𝑦𝑒𝑎𝑟)This allows us to predict plate diameter for a particular year.Substitute 2000 for year and we predict a diameter of (2000) = 10.8 in.
34Growing Plates?𝑦 =− 𝑥What is the predicted diameter for a plate manufactured in the year 2001?(2001) = in.How does this compare to our prediction for the year 2000 (10.8 in.)?largerSlope b = means that diameters are predicted to increase by inches per year on average
35Growing Plates?Slope is the predicted change in the response variable for one-unit change in the explanatory variable.Both the slope and the correlation coefficient for this study were positive.The slope and correlation coefficient will always have the same sign.
36Growing Plates?The y-intercept is where the regression line crosses the y-axis or the predicted response when the explanatory variable equals 0.We had a y-intercept of in the dinner plate equation. What does this tell us about our dinner plate example?Dinner plates in year 0 were inches.How can it be negative?The equation works well within the range of values given for the explanatory variable, but fails outside that range.Our equation should only be used to predict the size of dinner plates from about 1950 to 2010.
37Growing Plates?Predicting values for the response variable for values of the explanatory variable that are outside of the range of the original data is called extrapolation.
38Predicting Height from Footprints Exploration 10.3
39Inference for the Regression Slope: Theory-Based Approach Section 10.5
40Beta vs RhoTesting the slope of the regression line is equivalent to testing the correlation.Hence these hypotheses are equivalent.Ho: β = 0 Ha: β > 0 (Slope)Ho: ρ = 0 Ha: ρ > 0 (Correlation)Sample slope (b) Population (β: beta)Sample correlation (r) Population (ρ: rho)When we do the theory based test, we will be using the t-statistic which can be calculated from either the slope or correlation.
41IntroductionWe have seen in this chapter that our null distributions are again bell-shaped and centered at 0 (for either correlation or slope as our statistic). This is also true if we use the t-statistic.
42Validity ConditionsUnder certain conditions, theory-based inference for correlation or slope of the regression line use t-distributions.We are going to use the theory-based methods for the slope of the regression line. (The applet allows us to do confidence intervals for slope.)We would get the same p-value if we used correlation as our statistic.
43Validity ConditionsValidity Conditions for theory-based inference for slope of the regression equation.The values of the response variable at each possible value of the explanatory variable must have a normal distribution in the population from which the sample was drawn.Each normal distribution at each x-value must have the same standard deviation.
44Validity ConditionsSuppose the graph represents a scatterplot for an entire population where the explanatory variable is heart rate and the response is body temperature.Normal distributions oftemps for each BPMEach normal distributionhas the same SD
45Validity Conditions How realistic are these conditions? In practice, you can check these conditions by examining your scatterplot.Let’s look at some scatterplots that do not meet the requirements.
46Smoking and DrinkingThe relationship between number of drinks and cigarettes per week for a random sample of students at Hope College.The dot at (0,0)represents 524studentsAre conditions met?
47Smoking and DrinkingSince these conditions aren’t met, we shouldn’t apply theory-based inference
48When checking conditions for traditional tests use the following logic. Assume that the condition is met in your dataExamine the data in the appropriate mannerIf there is strong evidence to contradict that the condition is met, then don’t use the test.
49Validity ConditionsWhat do you do when validity conditions aren’t met for theory-based inference?Use the simulated-based approachAnother strategy is to “transform” the data on a different scale so conditions are met.The logarithmic scale is common
50Predicting Heart Rate from Body Temperature Example 10.5A
51Heart Rate and Body Temp Earlier we looked at the relationship between heart rate and body temperature with 130 healthy adults𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝐻𝑒𝑎𝑟𝑡 𝑅𝑎𝑡𝑒=− 𝑇𝑒𝑚𝑝r = 0.257
52Heart Rate and Body Temp We tested to see if we had convincing evidence that there is a positive association between heart rate and body temp in the population using a simulation-based approach. (We will make it 2-sided this time.)Null Hypothesis: There is no association between heart rate and body temperature in the population. β = 0Alternative Hypothesis: There is an association between heart rate and body temperature in the population. β ≠ 0
53Heart Rate and Body Temp We get a very small p-value (≈ 0.006) since anything as extreme as our observed slope of 2.44 is very rare
54Heart Rate and Body Temp We can also approximate a 95% confidence intervalobserved statistic + 2 SD of statistic 2.44 ± 2(0.850) = to 4.14What does this mean?We’re 95% confident that, in the population of healthy adults, each 1º increase in body temp is associated with an increase in heart rate of between 0.74 to 4.14 beats per minute
55Heart Rate and Body Temp The theory-based approach should work well since the distribution has a nice bell shapeAlso check the scatterplot
56Heart Rate and Body Temp Let’s do the theory-based test using the applet.We will use the t-statistic to get our theory-based p-value.We will find a theory-based confidence interval for the slope.After we are done with that, do Exploration 10.5: Predicting Brain Density from Facebook Friends