Subject: Psychopathy

Transcription

1 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 1: Research Skills: Statistics Problem Sheet 3: (Correlation and Regression): 1. The following numbers represent data from 1 individuals. Each individual has supplied a score on a test of psychopathy, and has also supplied a record of how many kitchen knives (think about it!) they have had to buy during the past year. Subject: Psychopathy score: o. knives bought: (a) Draw a scatterplot of these data. (b) Work out the Pearson's Correlation Coefficient for these data. What do you conclude? [Answer: r 0.86].. The following data represent data from students. Each student provided two scores: one is their performance on a statistics exam, and the other is a measure of their anxiety level the day before the exam. Subject: Statistics score: Anxiety score: (a) Draw a scatterplot of these data. (b) Work out the Pearson's correlation coefficient for these data. What do you conclude? [Answer: -.]. 3. Each of twelve motorcyclists provides data on the number of miles that they travel per year, and their rated level of happiness on a hundred-point scale (where 1 unhappy, and 0 ecstatic). Is there a relationship between the number of miles travelled and rated level of happiness? Motorcyclist: Miles travelled (in thousands): Happiness rating: (a) Draw a scatterplot of these data. (b) Calculate the Spearman's correlation coefficient for these data, and state your conclusions. [Answer: 0.968]. (c) Calculate the Pearson's correlation coefficient for these data. [Answer: 0.90]. Why does this answer differ from that for (b)?

2 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page : 4. For each of ten head-injury sufferers, a record is taken of the length of time spent in a coma after the accident, and the length of time taken before the patient returned to full-time work. Patient: coma length (hours): time off work (weeks): (a) Draw a scatterplot of these data. (b) Calculate the Pearson's correlation coefficient for these data. What are your conclusions? [Answer:.893]. (c) Work out the linear regression formula for predicting amount of time off work for any particular legnth of coma. [Answer: time off work * coma length]. Using this formula, how long would you predict that someone would be off work if they had been in a coma for (i) 3 hours [answer: 7.3 weeks]; (ii) 16 hours [answer: 1.9 weeks]; (iii) 48 hours [answer: 6.8 weeks]; (iv) 96 hours [answer: 47.5 weeks].

3 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 3: Research Skills: Worked Solutions to Problem Sheet 3 (Correlation and Regression): Question 1. This is the formula for Pearson's "r": r X XY ( X) ( Y) ( X) ( Y) Y (a) First, work out the various values needed (ΣX, ΣY, ΣXY, etc.). We have two independent variables: a measure of psychopathy and a measure of number of knives bought. Let's call the former the "X" variable, and the latter the "Y" variable. (We could equally well have called the psychopathy scores the "Y" values; it doesn't matter which way round we do it, as long as we are consistent throughout). Subject: Psychopathy o.knives score (X): bought (Y): X Y XY ΣX 89 ΣY 117 ΣX 949 ΣY 055 ΣXY 1311 (ΣX) (ΣY) (b) ow start working out the various bits of the equation: X ( X ) Y ( Y ) r

4 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 4: Our conclusion is that there is a strong positive correlation between a subject's psychopathy score ansd the number of knives purchased: the higher the psychopathy score, the more knives bought, and vice versa. Question. Subject : Statistics score (X) Anxiety score (Y) X Y XY ΣX 681 ΣY 691 ΣX ΣY 519 ΣXY 4678 (ΣX) (ΣY) X (b) ow start working out the various bits of the equation: ( X) Y ( Y) Our conclusion is that there is a very small (i.e., negligible) correlation between statistics scores and anxiety scores: the correlation is negative, suggesting that higher statistics scores are associated with lower anxiety scores, but there is essentially no relationship between performance on these two variables. Question 3. There are at least two different ways of calculating Spearman's rho. One way is to use the following formula: 6 rho 1 3 D To get ΣD and 3 -, you have to do the following: (a) rank the scores for each variable separately. Give the lowest score a rank of 1; the next lowest a score of ; and so on, until you have ranked all the scores for that variable. If two scores are the same, they are "tied". If there are two identical scores, give each of them the average of the two ranks which would have been assigned to them, had they been different numbers. Then give the next

5 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 5: score the rank which follows the ranks which have been averaged together. An example makes this clearer: score: rank: We had two identical scores, 40 ansd 40. Therefore they get the average of the ranks which they would have obtianed had they been different from each other (i.e., the average of the ranks "4" and "5", which is (4+5)/ 4.5). Although they both get a rank of 4.5, we regard the ranks of 4 and 5 as "used up" and give the next score which is different, a rank of 6. If we had had three identical scores of 40, the ranking would have gone like this: score: rank: The three scores of 40 are given the average of the ranks which they would have been given, had they been different scores (i.e., (4+5+6)/3 5). Ranks 4, 5 and 6 are now "used up"; so we continue with rank number 7. (b) Once the scores are ranked, subtract one set of ranks from the other (keeping a note of the signs of the differences, i.e., whther they are positive or negative). This gives you D. (c) Square these differences. This gives you D. (d) Add together the results of step (c). This gives you ΣD. (e) Multiply the result of step (d) by 6. (f) is the number of pairs of scores, and 3 is this number cubed. Subtract from 3. (g) Divide the result of step (e) by the result of step (f). (g) Finally, subtract the result of step (g) from 1. You should have your value of Spearman's rho. Another way of calculating Spearman's rho is to rank the scores for each variable in the way described above, and then use these ranks as the raw data in the Pearson's correlation formula. The following table shows the raw data for the X and Y variables; their ranks; D; and D. Subject: Miles Happiness travelled (X) rating (Y) ranks for X ranks for Y D D If you calculated Spearman's rho with the formula shown above, you get: D rho If you entered the ranks into the Pearson's correlation formula, you should have got: Subject Ranks for Ranks for

6 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 6: : miles (X) happiness X Y XY (Y) ΣX 78 ΣY 78 ΣX ΣY ΣXY 645 (ΣX) (ΣY) The rest of the calculations are... X ( X) Y ( Y) r Subject : Part (c): Here are the calculations for the Pearson's correlation coefficient: Miles travelled (X) Happines s rating (Y) X Y XY ΣX 60 ΣY 654 ΣX ΣY 4806 ΣXY 4960 (ΣX) (ΣY) The rest of the calculations are...

7 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 7: X ( X) Y ( Y) r The reason for the difference between the Spearman's and Pearson's correlation coefficients should have become apparent when you drew your scatterplot. Pearson's correlation coefficient is a measure of the linear relationship between two variables: the size of the correlation is reduced if they are not linearly related, even if the relationship is in other respects a strong one (as is the case here). Spearman's correlation coefficient is a measure of the degree of monotonic relationship between them; it simply measures the extent to which one of the two variables increases or decreases as a consequence of the other, largely independently of how much increase or decrease occurs. (Linear relationships are merely one type of monotonic relationship). The scatterplot shows that the relationship between the two variables is strong, but not perfectly linear. This affects the Pearson's correlation more than it affects the Spearman's correlation. Question 4. Patient: coma length time off (X) work (Y) X Y XY ΣX 19 ΣY 143 ΣX 6864 ΣY 793 ΣXY 41 (ΣX) (ΣY) The rest of the calculations...

8 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 8: X ( X) Y ( Y) r The conclusion is that there is a very strong positive correlation between length of coma and amount of time off work: the longer the coma, the lengthier the time off work, and vice versa. Part (c): Once you have the Pearson's correlation coefficient, it's very quick and easy to calculate the regression line, since the two formulae have a lot of bits in common. The formula for a straight line is: Y a + b * X First, work out b, the slope of the regression line: b XY X ( X) ( Y) ( X) b Second, work out a, the intercept of the regression line (i.e., the point at which the regression line intersects with the vertical axis of the graph): a Y b X Y a X 19.0 ( ) a

9 Research Skills Problem Sheet 3 : Graham Hole, March 009: Page 9: We now have the formula for drawing our regression line: Y' (0.433 * X) Y' means "predicted Y". All we have to do is replace the X in this formula with any value for which we want a predicted Y score. Thus, for an actual X score of 3, the predicted Y score is... Y' (0.433 * 3) For an actual X score of 16, the predicted Y score is... Y' (0.433 * 16) For an actual X score of 48, the predicted Y score is... Y' (0.433 * 48) For an actual X score of 96, the predicted Y score is... Y' (0.433 * 96) As checks to make sure that your regression equation is correct, you could try plotting the line to make sure that it does indeed go though most of your actual data points: if it is wildly off (especially if the correlation is a good one), then you have probably made a mistake in the calculations. Another check is to enter an X value for which you already have an actual Y value, and see if you get a predicted Y value which is reasonably similar to it. Thus, in this example, we have an actual X/Y data pair of hours coma and 16 weeks off work. If we enter this X value into our equation, we get a predicted Y value of.3 - not far off our actual Y value for that particular X value! (ote that both of these checks do presuppose that the correlation between the two variables is reasonably good in the first place; otherwise, predicted and actual Y values may be quite dissimilar).