Name:_____________________________ HP340: Statistical Methods, Fall 2016 Due Date: Nov 28, 2016 Homework #8 Instructions: Clearly circle the letter of the best answer for multiple-choice questions and show your work when calculations are required. Where necessary, please explain your reasoning for the approach you used. 1. Suppose two variables under study are temperature in degrees Fahrenheit (y) and temperature in degrees Celsius (x). The 'regression line' for this situation is: 9 Y^ = X +32 5 Assuming there is no error in observing temperature, the correlation coefficient would be expected to be (2 points) a. b. c. d. e. 2. 9 5 5 9 -1 +1 0 An investigator studies 50 pairs of unlike twins and reports that male birth weight (y) can be predicted by female birth weight (x) using the following equation (all weights in grams): Y^ =0.403 X + 1221 One can conclude from this that: a. b. c. d. e. (2 points) mean weight of twin brothers of girls who weigh 1000 g is predicted to be 1624 g mean weight of twin sisters of boys who weigh 1000 g is predicted to be 1624 g sample mean weight of the girls is 1221 g sample mean weight of the boys is 1221 g the correlation between girl's weight and boy's weight is 0.403 3. A study was conducted to investigate the relationship between the estriol level of pregnant women and subsequent height (in centimeters) of their children at birth. A Pearson's correlation coefficient was computed and found to be r = 0.411. The researcher decided to re-express height in millimeters rather than centimeters and then recomputed r. The recalculation yields the following value of r: (2 points) a. b. c. d. e. 4. Which equation best describes the regression line depicted in the figure on the right? (2 points) a. b. c. d. 5. 0.000 0.0411 0.00411 0.411 cannot be determined from data available Y^ =8 X +12 Y^ =8 X 12 ^ Y =8 X+ 12 Y^ =8 X12 The regression equation for predicting the number of speeding tickets from the driver's age is Y^ =0.06 X +5.5 . How many tickets do you predict a 20 year old would get? (2 points) a. 5.5 b. 5.0 c. 4.3 d. 0.06 e. -0.06 6. Given the following information calculate the Pearson's correlations between variable Jazz (x) and variable Hands (y). Using a two-tailed test at =.05, is the correlation coefficient significant? What is the percentage of variance in variable Hands that is associated with variance in variable Jazz (or vice versa)? (Write an informative and complete conclusion statement) (8 points) N=10; X =4.1; Y =4.7; s 2x = 6.77; s 2y = 1.34; CP = 22.3 7. Given the information from Question 6, what is the equation for the regression line with Jazz (x) predicting Hands (y)? (Write an informative and complete conclusion statement) (8 points) The SPSS file posted on BlackBoard (HW8.data.sav) contains data for the following questions. Imagine that you are researcher studying the impact of the number of friends on feelings of anxiety (on a scale of 0-50) following a cataclysmic world event. Use the SPSS file posted on BlackBoard (HW8.data.sav) to run a regression and answer the following questions. 8. Provide the SPSS output for the regression analysis. What is the equation for the regression line? Write a conclusion statement based on these results (be sure to include the statistical evidence concerning whether or not the slope is significant at =.05). (15 points) 9. Assume that a Low Anxiety score is equal to or less than 15. According to your regression equation, how may friends would you predict a person would need in order to have Low Anxiety? (5 points) 10. Imagine that an individual had no friends. What is her/his predicted anxiety score? (5 points) Extra Credit: What is the standard error of estimate? What is the standard error of the slope? (4 points) $FL2@(#) IBM SPSS STATISTICS 64-bit Macintosh 22.0.0.0 ################P#########Y@16 Nov 1611:00:04 ###########################SUBJECTI########################DRUGDOSE############# ###########FRIENDS ########################ANXIETY ############################################################# ############ ############################### ########### ##################D###SUBJECTI=SubjectID DRUGDOSE=DrugDose FRIENDS=Friends ANXIETY=Anxiety########################P###################V###SubjectID: $@Role('0' )/DrugDose:$@Role('0' )/Friends:$@Role('0' )/Anxiety:$@Role('0' ) #######ifnfkdmfjgmemdjgighheggdkgggifjdkiii jdhegjihkhng~gj}eg}mf}yni|gg|ig| jj{zinzfkylnlxgijxmhmwwfnv}llvjjvelluhkuifjupgjumju~kltlgtkhnsjhjskhshmr {glrhhrnenpsfmpjlpoknoinohlloilomkorjjoihoijnihnknm| kjmqlllxmllmnkmjkmhkvnljffjunnitflinlifgkiiiigjglgg \fHealth Behavior Statistical Methods HP 340L Lecture 15 Correlation Chapter 13 HP-340: Fall 2016 1 Last Lecture One-way ANOVA Partitioning the variance Hypothesis testing using the F-test Post-hoc analysis using Tukey's HSD test HP-340: Fall 2016 2 Today's Lecture Scatterplots Correlation Review of Pearson's Correlation Spearman's Correlation Characteristics of correlation Direction, degree of relationship, coefficient of determination, causality Statistical significance Hypothesis test H0: = 0 vs. H1: 0 HP-340: Fall 2016 3 Introduction to Correlation 2-sample t-test and ANOVA can be used in designed experiments to learn about the relationship between: Independent variable(s): categorical/nominal (manipulated) Dependent variable: quantitative (measured, e.g., background music test scores) Correlational studies ask whether 2 variables are related when: Both variables continuous Neither variable is manipulated No dependent or independent variables Direction and strength of relationship is of interest HP-340: Fall 2016 4 Scatterplot Every subject is represented as a single point on the (X,Y) graph Covariation: Two variables co-vary when a change in one (X) is related to a change in the other (Y) HP-340: Fall 2016 5 Scatterplot Linear relationship: relationship between two variables that can be approximated by a straight line In a perfect linear relationship, knowing the score on one variable allows us to exactly predict the other variable score HP-340: Fall 2016 6 Scatterplot 1 - linear Scatterplot 2 - perfect Positive [direct] relationship: As the value of X increases, the value of Y increases Y = Smoke Scatterplot Negative [inverse] relationship: As the value of X increases, the value of Y decreases Y = Vision X = Drink X = Age HP-340: Fall 2016 7 What is Correlation Correlation captures the extent to which two variables have a linear relationship. To calculate correlations we need pairs of numbers, i.e., height =60 inches, weight=150 pounds Correlation coefficients \"r\" are descriptive statistics that describe the degree/strength of relationship between 2 variables. HP-340: Fall 2016 8 Pearson Correlation Coefficient Karl Pearson (1857 - 1936) came up with a number to represent this relationship and called it Pearson Product Correlation. Pearson correlation indicates the degree of linear relationship between 2 variables measured at the interval or ratio level. HP-340: Fall 2016 9 The Pearson Correlation Coefficient The Pearson correlation coefficient (r): HP-340: Fall 2016 10 The Pearson Correlation Coefficient The standard error of the correlation coefficient (sr) provides a measure of the precision in the estimate of r Computed as: Recall the correlation coefficient is unitless HP-340: Fall 2016 11 The Pearson Correlation Coefficient The numerator of r is the covariance, Also called the \"cross products\" The denominator includes the standard deviations of X and Y HP-340: Fall 2016 12 The Pearson Correlation Coefficient The standard scores formula for the correlation coefficient Standardized scores (zX and zY) SX and SY are the sample SDs Npairs is the number of pairs of scores HP-340: Fall 2016 13 The Pearson Correlation Coefficient Sample SDs Note for the sample SDs you divide by only N HP-340: Fall 2016 14 The Pearson Correlation Coefficient Strength of a correlation |r| = 1.0 HP-340: Fall 2016 Perfect |r| = 0.8 - 0.9 Great |r| = 0.6 - 0.79 Good |r| = 0.4 - 0.59 Moderate |r| = 0.2 - 0.39 Poor |r| = 0.09 - .019 Weak |r| = 0 None 15 The Pearson Correlation Coefficient Correlation examples: Height & weight among college students Height and body temperature Age and resting body temperature Abstract reasoning and verbal reasoning 0.92 0 -0.15 0.67 Some uses of correlation Matched pairs IQs of wives and husbands Criterion validity Correlating scores of one test with scores of another test, e.g., ACT and SAT scores should correlate Predictive validity Correlating scores of a test with future outcomes, e.g., GRE scores predict completion of graduate school HP-340: Fall 2016 16 The Pearson Correlation Coefficient Example: A psychologist has 10 patients to whom he administers two performance tests, visual and hearing The following are the scores of the 10 patients on those two tests Patient: A Visual : 15 Hearing : 20 HP-340: Fall 2016 B 12 15 C 10 12 D 14 18 E 10 10 17 F 8 13 G 6 12 H 15 10 I 16 18 J 13 15 The Pearson Correlation Coefficient Scatterplot of the data HP-340: Fall 2016 18 The Pearson Correlation Coefficient Results HP-340: Fall 2016 19 The Pearson Correlation Coefficient Effect of outliers Create an outlier in the example HP-340: Fall 2016 Original Data Data with Outlier r = 0.568 r = 0.276 20 The Pearson Correlation Coefficient Hypothesis testing: are X and Y linearly associated? Forms of hypothesis: H0: H0: H0: H0: = 0; not linearly associated < 0; negatively linearly associated > 0; positively linearly associated 0; linearly associated Test statistic: HP-340: Fall 2016 21 The Pearson Correlation Coefficient Remember: r is a statistic and is subject to random variation r is an estimate of the true population correlation () In the statement of the null and alternative, you are stating whether X and Y are correlated or not in the population H0: X and Y are not correlated in the population H1: X and Y are correlated in the population HP-340: Fall 2016 22 The Pearson Correlation Coefficient Alternative to the t-statistic approach Calculate robs Degrees of freedom: N - 2 For critical values use Table C5 Identify df and -level to obtain r critical value If |robs| > rcrit then reject the null hypothesis HP-340: Fall 2016 23 The Pearson Correlation Coefficient robs = 0.57, N = 10 subjects df : N - 2 = 8 Critical value from Table C5: rcrit = 0.632 |robs| = 0.57 < 0.632 = rcrit then we DO NOT reject the null hypothesis HP-340: Fall 2016 24 The Coefficient of Determination Symbolized by \"r2\" - simply square r Similar to 2 Indicates the common variance of variables X and Y It is the proportion of variance in Y that is explained by X (strength of the relationship) If the correlation between visual and hearing scores is 0.568, the coefficient of determination is: r2 = 0.5682 = 0.322624 35% of the variability in visual score can be explained by hearing score or vice versa HP-340: Fall 2016 25 The Spearman Rank Correlation A different measure of relationship strength This correlation can be used with: Data that are not normally distributed Ordinal data Robust to outliers Captures monotonic relationship between X & Y (not necessarily linear) HP-340: Fall 2016 26 The Spearman Rank Correlation Rank the X values Rank the Y values If you have ties, take the average of the ranks D = the difference of the ranks for each X, Y pair 6 D 2 rS = 1 2 N pairs ( N pairs 1) HP-340: Fall 2016 27 The Spearman Rank Correlation Do happier people have more friends? A researcher tested people with a happiness scale (0 - 30). They then ranked each of the 8 people based on the number of friends they said they had (1 = most friends). What is the Spearman rank correlation? HP-340: Fall 2016 28 The Spearman Rank Correlation The data Happiness score 25 18 29 13 25 19 30 24 18 19 28 Happiness Score (ranked) 4.5 9.5 2 11 4.5 7.5 1 6 9.5 7.5 3 Number of friends (ranked) 6 8 4.5 10 2 10 3 7 4.5 10 1 D -1.5 1.5 -2.5 1 2.5 -2.5 -2 -1 5 -2.5 2 D2 4.5 9.5 2 11 4.5 7.5 1 6 9.5 7.5 3 D 2 = 64.5 HP-340: Fall 2016 29 The Spearman Rank Correlation Compute rS 6 D 2 rS = 1 2 N pairs ( N pairs 1) 6 ( 64.5 ) =1 11(11 1) = 0.7068 Test for significance in the same manner as Pearson correlation HP-340: Fall 2016 30 The Spearman Rank Correlation Example: What is the relationship between playing and chasing behavior of cats? Animals are ranked on both playing and chasing behavior Cats 1 - 10 were observed and ranked on playing and chasing behaviors The most playful cat receives a rank of 1 and the least playful receives a rank of 10 HP-340: Fall 2016 31 The Spearman Rank Correlation Ranked data for playing (X) and chasing (Y) Note: Cat #1 and #10 received a tie in ranking, so theirrank of 6.5 is the average of ranks 6 and 7 HP-340: Fall 2016 32 Chasing Rank 1 Playing Rank 2 2 6 8 3 8 4 4 4 1 5 7 5 6 5 9 7 10 10 8 3 2 9 1 3 10 9 6.5 Cat 6.5 The Spearman Rank Correlation HP-340: Fall 2016 33 The Spearman Rank Correlation rS = 0.511 Is this correlation statistically significant? Can test for significance using the t-statistic just like the Pearson's correlation Can also use Table C.6 from the textbook For = 0.05 and Npairs = 10, rCrit = 0.648 What do you conclude about H0: s = 0? HP-340: Fall 2016 34 The Spearman Rank Correlation When you select Spearman's correlation, summary statistics are not available Only get the correlation output What conclusion do you make? HP-340: Fall 2016 35 The Spearman Rank Correlation The Spearman correlation is appropriate for continuous X and Y when the relationship is monotonic, but not linear rS = 0.30 r = 0.23 Pearson's correlation measures linear relationships HP-340: Fall 2016 36 \fHealth Behavior Statistical Methods HP 340L Lecture 15 Correlation Chapter 13 HP-340: Fall 2016 1 Last Lecture One-way ANOVA Partitioning the variance Hypothesis testing using the F-test Post-hoc analysis using Tukey's HSD test HP-340: Fall 2016 2 Today's Lecture Scatterplots Correlation Review of Pearson's Correlation Spearman's Correlation Characteristics of correlation Direction, degree of relationship, coefficient of determination, causality Statistical significance Hypothesis test H0: = 0 vs. H1: 0 HP-340: Fall 2016 3 Introduction to Correlation 2-sample t-test and ANOVA can be used in designed experiments to learn about the relationship between: Independent variable(s): categorical/nominal (manipulated) Dependent variable: quantitative (measured, e.g., background music test scores) Correlational studies ask whether 2 variables are related when: Both variables continuous Neither variable is manipulated No dependent or independent variables Direction and strength of relationship is of interest HP-340: Fall 2016 4 Scatterplot Every subject is represented as a single point on the (X,Y) graph Covariation: Two variables co-vary when a change in one (X) is related to a change in the other (Y) HP-340: Fall 2016 5 Scatterplot Linear relationship: relationship between two variables that can be approximated by a straight line In a perfect linear relationship, knowing the score on one variable allows us to exactly predict the other variable score HP-340: Fall 2016 6 Scatterplot 1 - linear Scatterplot 2 - perfect Positive [direct] relationship: As the value of X increases, the value of Y increases Y = Smoke Scatterplot Negative [inverse] relationship: As the value of X increases, the value of Y decreases Y = Vision X = Drink X = Age HP-340: Fall 2016 7 What is Correlation Correlation captures the extent to which two variables have a linear relationship. To calculate correlations we need pairs of numbers, i.e., height =60 inches, weight=150 pounds Correlation coefficients \"r\" are descriptive statistics that describe the degree/strength of relationship between 2 variables. HP-340: Fall 2016 8 Pearson Correlation Coefficient Karl Pearson (1857 - 1936) came up with a number to represent this relationship and called it Pearson Product Correlation. Pearson correlation indicates the degree of linear relationship between 2 variables measured at the interval or ratio level. HP-340: Fall 2016 9 The Pearson Correlation Coefficient The Pearson correlation coefficient (r): HP-340: Fall 2016 10 The Pearson Correlation Coefficient The standard error of the correlation coefficient (sr) provides a measure of the precision in the estimate of r Computed as: Recall the correlation coefficient is unitless HP-340: Fall 2016 11 The Pearson Correlation Coefficient The numerator of r is the covariance, Also called the \"cross products\" The denominator includes the standard deviations of X and Y HP-340: Fall 2016 12 The Pearson Correlation Coefficient The standard scores formula for the correlation coefficient Standardized scores (zX and zY) SX and SY are the sample SDs Npairs is the number of pairs of scores HP-340: Fall 2016 13 The Pearson Correlation Coefficient Sample SDs Note for the sample SDs you divide by only N HP-340: Fall 2016 14 The Pearson Correlation Coefficient Strength of a correlation |r| = 1.0 HP-340: Fall 2016 Perfect |r| = 0.8 - 0.9 Great |r| = 0.6 - 0.79 Good |r| = 0.4 - 0.59 Moderate |r| = 0.2 - 0.39 Poor |r| = 0.09 - .019 Weak |r| = 0 None 15 The Pearson Correlation Coefficient Correlation examples: Height & weight among college students Height and body temperature Age and resting body temperature Abstract reasoning and verbal reasoning 0.92 0 -0.15 0.67 Some uses of correlation Matched pairs IQs of wives and husbands Criterion validity Correlating scores of one test with scores of another test, e.g., ACT and SAT scores should correlate Predictive validity Correlating scores of a test with future outcomes, e.g., GRE scores predict completion of graduate school HP-340: Fall 2016 16 The Pearson Correlation Coefficient Example: A psychologist has 10 patients to whom he administers two performance tests, visual and hearing The following are the scores of the 10 patients on those two tests Patient: A Visual : 15 Hearing : 20 HP-340: Fall 2016 B 12 15 C 10 12 D 14 18 E 10 10 17 F 8 13 G 6 12 H 15 10 I 16 18 J 13 15 The Pearson Correlation Coefficient Scatterplot of the data HP-340: Fall 2016 18 The Pearson Correlation Coefficient Results HP-340: Fall 2016 19 The Pearson Correlation Coefficient Effect of outliers Create an outlier in the example HP-340: Fall 2016 Original Data Data with Outlier r = 0.568 r = 0.276 20 The Pearson Correlation Coefficient Hypothesis testing: are X and Y linearly associated? Forms of hypothesis: H0: H0: H0: H0: = 0; not linearly associated < 0; negatively linearly associated > 0; positively linearly associated 0; linearly associated Test statistic: HP-340: Fall 2016 21 The Pearson Correlation Coefficient Remember: r is a statistic and is subject to random variation r is an estimate of the true population correlation () In the statement of the null and alternative, you are stating whether X and Y are correlated or not in the population H0: X and Y are not correlated in the population H1: X and Y are correlated in the population HP-340: Fall 2016 22 The Pearson Correlation Coefficient Alternative to the t-statistic approach Calculate robs Degrees of freedom: N - 2 For critical values use Table C5 Identify df and -level to obtain r critical value If |robs| > rcrit then reject the null hypothesis HP-340: Fall 2016 23 The Pearson Correlation Coefficient robs = 0.57, N = 10 subjects df : N - 2 = 8 Critical value from Table C5: rcrit = 0.632 |robs| = 0.57 < 0.632 = rcrit then we DO NOT reject the null hypothesis HP-340: Fall 2016 24 The Coefficient of Determination Symbolized by \"r2\" - simply square r Similar to 2 Indicates the common variance of variables X and Y It is the proportion of variance in Y that is explained by X (strength of the relationship) If the correlation between visual and hearing scores is 0.568, the coefficient of determination is: r2 = 0.5682 = 0.322624 35% of the variability in visual score can be explained by hearing score or vice versa HP-340: Fall 2016 25 The Spearman Rank Correlation A different measure of relationship strength This correlation can be used with: Data that are not normally distributed Ordinal data Robust to outliers Captures monotonic relationship between X & Y (not necessarily linear) HP-340: Fall 2016 26 The Spearman Rank Correlation Rank the X values Rank the Y values If you have ties, take the average of the ranks D = the difference of the ranks for each X, Y pair 6 D 2 rS = 1 2 N pairs ( N pairs 1) HP-340: Fall 2016 27 The Spearman Rank Correlation Do happier people have more friends? A researcher tested people with a happiness scale (0 - 30). They then ranked each of the 8 people based on the number of friends they said they had (1 = most friends). What is the Spearman rank correlation? HP-340: Fall 2016 28 The Spearman Rank Correlation The data Happiness score 25 18 29 13 25 19 30 24 18 19 28 Happiness Score (ranked) 4.5 9.5 2 11 4.5 7.5 1 6 9.5 7.5 3 Number of friends (ranked) 6 8 4.5 10 2 10 3 7 4.5 10 1 D -1.5 1.5 -2.5 1 2.5 -2.5 -2 -1 5 -2.5 2 D2 4.5 9.5 2 11 4.5 7.5 1 6 9.5 7.5 3 D 2 = 64.5 HP-340: Fall 2016 29 The Spearman Rank Correlation Compute rS 6 D 2 rS = 1 2 N pairs ( N pairs 1) 6 ( 64.5 ) =1 11(11 1) = 0.7068 Test for significance in the same manner as Pearson correlation HP-340: Fall 2016 30 The Spearman Rank Correlation Example: What is the relationship between playing and chasing behavior of cats? Animals are ranked on both playing and chasing behavior Cats 1 - 10 were observed and ranked on playing and chasing behaviors The most playful cat receives a rank of 1 and the least playful receives a rank of 10 HP-340: Fall 2016 31 The Spearman Rank Correlation Ranked data for playing (X) and chasing (Y) Note: Cat #1 and #10 received a tie in ranking, so theirrank of 6.5 is the average of ranks 6 and 7 HP-340: Fall 2016 32 Chasing Rank 1 Playing Rank 2 2 6 8 3 8 4 4 4 1 5 7 5 6 5 9 7 10 10 8 3 2 9 1 3 10 9 6.5 Cat 6.5 The Spearman Rank Correlation HP-340: Fall 2016 33 The Spearman Rank Correlation rS = 0.511 Is this correlation statistically significant? Can test for significance using the t-statistic just like the Pearson's correlation Can also use Table C.6 from the textbook For = 0.05 and Npairs = 10, rCrit = 0.648 What do you conclude about H0: s = 0? HP-340: Fall 2016 34 The Spearman Rank Correlation When you select Spearman's correlation, summary statistics are not available Only get the correlation output What conclusion do you make? HP-340: Fall 2016 35 The Spearman Rank Correlation The Spearman correlation is appropriate for continuous X and Y when the relationship is monotonic, but not linear rS = 0.30 r = 0.23 Pearson's correlation measures linear relationships HP-340: Fall 2016 36 \f\f