HUMBEHV 3HB3 Correlation Week 3

Transcription

1 HUMBEHV 3HB3 Correlation Week 3 Prof. Patrick Bennett Correlation & Regression Over the net two weeks, we discuss two concepts that focus on the relationship between variables: correlation and regression Correlation: a measure of the linear association between 2 variables Regression: estimates the best-fitting line that relates a predictor variable,, to a criterion variable,. - used to understand how changes when varies Correlation of Variables Correlations in Research Correlation can be calculated for: - dependent variable (i) and independent variable (i) - 2 dependent variables (1i and 2i) - 2 independent variables (1i and 2i) Eamples from hpothetical stud eamining antidepressants & sleep: - i = mood state, i = antidepressant dose - 1i = sleep qualit, 2i = time slept - 1i = antidepressant dose, 2i = tpe of depression effectiveness of fluoridated water in reducing cavities association between smoking and lung cancer apparentl false claim of link between autism and measles, mumps, and rubella (MMR) inoculations proposed that MMR causes autism drinking coffee associated with longer life claim that orchestra conductors live longer

2 Eploring data so far Scatterplot 4 i 4 i Regression line: the line of best fit ; represents best prediction of i for a given value of i 3 3 Frequenc 2 Frequenc 2 (i, i) = (16, 2) Phsicians Infant Mortalit Correlation: Is there are a relationship between these 2 variables? To generate a scatterplot, ou need the actual matched pairs of the values of i and i. Correlation strength & direction Negative Positive Correlation depends on fit, not slope Strong Weak r = r = r 0.8 r = 0.8 r r = r = 0.63

3 Perfect Linear Relation (r = 1 or r = -1) No linear association (r = 0) perfect positive correlation perfect negative correlation r = 0 r = r = 1.0 r = Correlations & deviations around the means r = 0.7 r = -0.7 Linear vs. curvilinear relations Monotonic Monotonic Monotonic Linear Non-linear Non-linear Non-monotonic Non-linear Dotted lines indicate means of and. Note that points do not fall within each quadrant equall. Monotonic: as increases, increases or decreases without reversal might not be a straight line! Non-monotonic: as increases, changes direction at least once Linear relationship: best fit line is straight Curvilinear: best fit line is not straight Correlation is onl sensitive to linear relations!

4 4 (,) data sets plotted with a reference line A B The correlation is the same in all 4 sets! Question: which set has the highest correlation? C D Correlation is a measure of linear association and is insensitive to non-linear association Covariance Statistic representing the degree to which 2 variables var together i= 1 cov = N (i - )(i - ) N -1 - where N is the number of observations. Note that covariance is the sum of products of deviation scores N (i - )(i - ) i= 1 cov = N -1 r = 0.7 Ȳ r = -0.7 Ȳ Green points drive COV in positive direction. Blue points drive COV in negative direction. Net COV differs from zero.

5 Correlation Coefficient, r N (i - )(i - ) i= 1 cov = N -1 Ȳ r = 0 Green points drive COV in positive direction. Blue points drive COV in negative direction. Note that net COV is zero. r = 0 Ȳ cov r = s s - s and s are the standard deviations of and respectivel. r is the Pearson Product-Moment Correlation Coefficient Note that r has no units Standardizes the covariance so that we can compare correlations even between groups with ver different dispersions Pearson r r ranges from -1 to 1 - r = +1 when and are perfectl positivel correlated - r = -1 when and are perfectl negativel correlated - r = 0 when there is no linear relationship between and in Pscholog, - Weak r: Moderate r: Strong r: >0.5 What does r mean? Measures magnitude (strength) & direction of linear relationship between 2 variables r = -1 r = +1 r = -0.6 r = +0.3 Q1. Does this mean there is 30% relationship between and? r = 0 r = 0 Q2. But, this looks like a perfect correlation? r is an inde of how much uncertaint about is reduced b knowing

6 r varies across samples Confidence Interval Frequenc Sample r (n = 20; true-r = 0.4) population r = 0.4 mean (sample r) = 0.39 range (sample r) = [-0.45, 0.88] Each sample r is an estimate of the population r Some estimates are good, others are not so good. Can we quantif the uncertaint of our estimate? A confidence interval is a range of values, calculated from the sample observations, that are believed, with a particular probabilit, to contain the true population parameter. A 95% confidence interval, for eample, implies that were the estimation process repeated again and again, then 95% of the calculated intervals would be epected to contain the true parameter value. Note that the stated probabilit level refers to the properties of the interval and not to the parameter itself which is not considered a random variable. B.S. Everitt, Dictionar of Statistics 95% Confidence Interval a range (interval) for a statistic (e.g., r) calculated from our sample the interval varies across samples in the long run, the interval contains the true population value 95% of the time eample: the sample correlation r was 0.37 (CI95% = [-0.08, 0.70]) Sample r Other Tpes of Correlations Two continuous variables - Pearson Product-Moment Correlation Coefficient (r) Two ranked/ordinal variables - Spearman s correlation coefficient for ranked data (rho (ρ) or rs) One dichotomous & one continuous variable - e.g., correct/incorrect answer on mc question and eam total score - Point-biserial Correlation (rpb): Calculate Pearson s r but call it rpb Two dichotomous variables - e.g. age (teens/seniors) and coin purse ownership (es/no) - Calculate Pearson s r but called in rφ Correlations with ranked data Spearman s correlation coefficient, rs - useful when observations have been replaced b their numerical ranks e.g., changing applicants eam scores to ranks english math english.rank math.rank

7 Spearman s rank order correlation Math r = Math Rank r s = rs is more robust to etreme high-leverage points r = r s = r = 0.38 r s = English English Rank Same as Pearson r correlation for ranks, not actual (,) values. Wh use rs if ou have the original values? Changing onl 2 out of 100 points markedl affects r but not rs but not to all tpes of etreme scores Alcohol Ependiture High Leverage Point (Figure 9.3) r = 0.22 r = 0.78 r s = 0.37 r s = 0.83 Northern Ireland Correlations with ranked data Spearman s correlation coefficient, rs, calculated with the same formula that is used to calculate Pearson s r - formula applied to ranks, not actual (,) values rs is much more resistant/robust than r to high leverage data points However, interpretations of r and rs differ: - r is an inde of the strength/direction of linear relation - rs is an inde of the strength/direction of monotonic relation Tobacco Ependiture

8 rs measures monotonicit of, association r = 0.58 r s = r = rs measures linear association between, ranks rs measures monotonic association between, scores r s = rs is sensitive to etreme scores affecting monotonicit Alcohol Ependiture High Leverage Point (Figure 9.3) r = 0.22 r = 0.78 Northern Ireland Tobacco Ependiture r s = 0.37 r s = 0.83 Alcohol Ependiture Reduction of Monotonicit Northern Ireland Tobacco Ependiture inclusion of Northern Ireland data point significantl alters monotonicit of the, association Other Tpes of Correlations Two continuous variables - Pearson Product-Moment Correlation Coefficient (r) Two ranked/ordinal variables - Spearman s correlation coefficient for ranked data (rho (ρ) or rs) One dichotomous & one continuous variable - e.g., correct/incorrect answer on mc question and eam total score - Point-biserial Correlation (rpb): Calculate Pearson s r but call it rpb Two dichotomous variables - e.g. age (teens/seniors) and coin purse ownership (es/no) - Calculate Pearson s r but called in rφ (phi) Point-biserial Correlation rpb eample variable: answers for a single multiplechoice question were coded as incorrect (0) or correct (1) variable: total score on remaining eam questions Correlate & : rpb = 0.46 What happens if we code incorrect and correct responses with other numbers (e.g., -10 & 10)? - magnitude rpb is not changed b coding scheme for variable (sign of rpb can change) eam score r pb = (incorrect) 1 (correct) item response

9 Other Tpes of Correlations Two continuous variables - Pearson Product-Moment Correlation Coefficient (r) Two ranked/ordinal variables - Spearman s correlation coefficient for ranked data (rho (ρ) or rs) One dichotomous & one continuous variable - e.g., correct/incorrect answer on mc question and eam total score - Point-biserial Correlation (rpb): Calculate Pearson s r but call it rpb Two dichotomous variables - e.g., gender (male/female) and correct/incorrect answer on mc question - Calculate Pearson s r but called in rφ (phi) Correlations with ranked data Source: Skbkekas on Wikipedia Correlations with ranked data It makes sense to investigate the correlation between English (mark) and Maths (mark) using r, because the data is on a ratio scale and the intervals are meaningful It makes sense to use rs for investigating the relationship between Rank (English) and Rank (Maths) because the data are ranked on an ordinal scale and the invervals are meaningless Factors that affect correlation Nonlinearit Etreme observations Range restrictions Heterogeneous subsamples

10 Restricted Range Restricted Range Weight of Brain b Age r = r = r = r = Brain Weight (grams) Age (ears) FIGURE 8. Increase in weight of brain b age. Reprinted with permission from Copoletta JM, Wolbach JB. Bod length and brain weight of infants and children. Am J Pathol. 1933;9: Restricted range can obscure a linear association. Restricted range can obscure a curvilinear association. Heterogeneous Subsamples Simpson (1951): - a statistical association observed in a population could be attenuated and even reversed within subgroups that make up that population Correlations calculated on populations consisting of heterogeneous subgroups ma be misleading Simpson s Parado Correlation for population is positive Correlations within sub-groups are negative Simpson's parado in pschological science: a practical guide Kievit, Frakenhuis, Waldorp, & Borsboom, Front. Pschol., 12 August 201

11 Simpson s Parado Correlation Causation Engineering PhDs & Mozzarella Consumption ( ) Overall correlation is approimatel zero Strong, opposite correlations in 2 sub-groups Simpson's parado in pschological science: a practical guide Kievit, Frakenhuis, Waldorp, & Borsboom, Front. Pschol., 12 August 2013 Mozzarella Cheese Consumption (lbs per capita) r = Civil Eng Doctorates Awarded Correlation Causation reading skill correlated with shoe size over last 2 centuries, price of bread in Great Britain is correlated with sea level in Venice rpb: BMW owners have higher incomes than Ford owners number of pirates is correlated with global temperature Global Average Temperature, C Pirates Cause Global Warming Global Average Temperature vs. Number of Pirates Number of Pirates (Approimate) Avg Global Temperature (C) r = Number of Pirates (Approimate) 1860 Source: Wikipedia

12 Correlation Causation cont d 1. The relationship could be causal. - Sunlight could protect against breast cancer (because it helps produce vitamin D) 2. The relationship ma be reversed. - Happiness could lead to more social relationships, or the other wa around. 3. The relationship could be partl causal. - Increased wealth could lead to increased happiness (if a supportive famil is also present) 4. Variables ma be changing over time. - Pirates and global warming are associated because both change over time 5. There ma be other confounding factors. - reading skill correlated with shoe size because both are correlated with age - wine consumption is associated with decrease in heart disease wine consumption is greater in sunn climates, and heart disease is lower in sunn regions perhaps wine/heart disease association reall is sun-eposure/heart disease association? 6. Both ma be other causal factors. - reading skill correlated with shoe size because both correlated with ears of education - Famil stabilit and phsical illness ma be related because stress is related to both. 7. Correlation ma be a coincidence (due to chance). Correlations (summar) Correlations: inde of the association between 2 variables Pearson r: inde of strength & direction of linear association - r = COV/(s s); varies between -1 & +1 - a measure of how much our uncertaint about the value of i is reduced b knowing the value of i - r is an estimate of the population correlation that varies across samples - a 95% Confidence Interval of r, which varies from sample to sample, contains the true population parameter 95% of the time Spearman s rs: inde of monotonicit of association - equivalent to Pearson s r for ranked data - generall more robust than r to etreme (i.e., high-leverage ) data points Correlations affected b: nonlinearit, etreme scores, range restrictions, heterogeneous samples CORRELATION CAUSATION