Z-SCORES AND CORRELATION! LECTURE#3! PSYC218 ANALYSIS OF BEHAV. DATA! DR. OLLIE HULME, 2011, UBC!

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1 Z-SCORES AND CORRELATION! LECTURE#3! PSYC218 ANALYSIS OF BEHAV. DATA! DR. OLLIE HULME, 2011, UBC!

2 Housekeeping! All assignments in-class hard copy only SPSS should be installed, last chance to report problems before assignment due. Mac patch unavailable, but only helps with graphs not required for assignments. Assignment 2 released tomorrow, based on survey monkey data Coglab Memory span due Thursday noon Assignment 1 due Thursday 5pm Assignment 2 due Tuesday 5 th July 5pm

3 Roadmap! Finish Z-scores Intro to correlation Scatter plots Straight line equations Correlation coefficients Correlation and causation

4 Z-score table Appendix D! Column A lists all of the various possible z scores Note it only lists positive scores. because the normal curve is symmetrical so the corresponding values for ve scores are identical. Column B lists the proportion of scores that fall between the z score (listed in column A) and the mean Column C lists the proportion of scores that fall between the zscore (listed in column A) and the closest tail of the distribution For positive z scores it gives the proportion of scores that are higher than the z score, For negative z scores it is the proportion that are lower

5 Question! You look up a z score of in Table A. Column C shows that.3085 corresponds to that z-score. This means that: a) 30.85% of the scores in the distribution are lower than the z score b) 30.85% of the scores intrick! the distribution are higher than the z score c) 30.85% of the scores and the z score d) Because the z-score table is positive only, you look lieupbetween mean the z-scorethe for 0.50, If are above this zscore, then for -0.5 the 80.85% of the scores insame the distribution proportion mustare be higher than the z score below Therefore 30.85% are e) 80.85% of the scores inlower the than distribution the z-scoreare lower than the z score If X percent are above a certain z-value, then the same percentage must be below the negative of that z-value

6 Percentile Points! If the memory test was given to an entire population and = 7 and = 1.88 Percentile point for 75% What is the score below which 75% of the scores fall? What is P75? 25% Step 1: Using Table A locate the area in Column C closest to (25%) and find its z-score Area closest = z value = Z Step 2: Transform z =.67 to a raw score. X = (z)( ) + X = (.67)(1.88) + 7 X = 8.26 So 75% of the population received a memory test score lower than 8.26

7 Further Illustration! If the memory test was given to an entire population and = 7 and = 1.88 What are the scores that bound (that define the boundary) the middle 90% of the distribution. 90% 5% 5% Z

8 Further Illustration! Step 1: Using Table A locate the area in column C closest to.0500 (5%) Because you want to know the score for which 5% of scores are higher and the score for which 5% of scores are lower as these scores will bound the middle 90%. (100-90)/2 = 5% find the corresponding z-score z=1.65 The other z-score will be because both boundaries are the same distance from the mean 90% 5% 5% Z Step 2: x = 3.90 x = Transform z = 1.65 and z = to raw scores via X = (z)( ) + ] z = 1.65 x = (1.65)(1.88) + 7 x = z = x = (-1.65) (1.88) + 7 x = 3.90 The scores 3.90 and bound the middle 90% of the distribution

10 Relations between variables! So far everything has been about individual distributions of data Often we want to know how one variable relates to another variable. This can help us predict one variable just by knowing the other. Z e.g. how does IQ relate to exam scores do those with higher scores perform better?. e.g. IQ, exam scores If there was a strong relationship between IQ and exam scores then knowing someone s IQ might be a good predictor of exam scores

11 Correlation! Provides means to assess the magnitude and direction of relationship between two variables Although it does not prove causality it is a first step toward investigating how one variable causes another If one variable correlates with another, then it might be causally related If there is no correlation then this is evidence against a causal relationship. which can then be tested experimentally with a true experiment) But you may not have enough data

12 Uses of correlation! Correlations are commonly calculated in observational studies but also in true experiments In some cases it is not practical or ethical to manipulate a variable and correlation is all we can do Assessing the reliability and validity of a measure often involves correlation (e.g., is marijuana use related to memory? how does income relate to childhood behavior?) (e.g. test-retest reliability, interrater reliability)

14 Scatter Plots! = a graph of paired X and Y values X = 7, Y=6 X = 2, Y=3 Each point represents a pair of X and Y values This can be plotted easily with SPSS, or any spreadsheet software Those with certain version of Mac OSX may not be able to do this in SPSS

15 Linear Relationships! A linear relationship between 2 variables is one in which the relationship can be most accurately represented by a straight line 7 Family Number of Children (X Variable) Adams 1 2 Brown 2 3 Carey 3 4 Davidson 4 5 Eastman 5 6 Average noise (Y variable) Noise level Number of children

17 Straight line equation! straight lines can represented by a simple equation 7 Equation of a straight line: 6 Y = bx + a 5 4 Noise level Number of children

18 Finding the Slope (b)! Y = bx + a 7 b = slope 6 a = Y intercept 5 (value of Y when X =0) 4 2,3 3,4 4,5 5,6 3 1, The slope is constant, therefore can be calculated from anywhere on the line

19 Question! What does b =.70 mean? a) For every 1 unit increase in Y, X increases by.70 b) For every 1 unit increase in X, Y increases by 0.70 c) For every.70 increase in X, Y increases by 1 unit d) B and C Way to remember this is slope = delta Y / delta X, therefore it is change in Y per unit change in X *Note the error on Page 107. In the text example the slope is.40. The text says Y increases by 1 unit for every.40 increase in X. This is incorrect it should say: Y increases by.40 units for every 1 unit increase in X.

20 Predicting Y Given X! Jon and Kate have 8 (X=8) children. Use the formula to determine what the average noise is (decibels) Y = bx + a a = Y intercept (value of Y when X =0) = 1 We know these values from the graph intercept b = slope of the line = 1 We know this from our slope calculation Y = 1 X + 1 Y' = 1(8)+1 = 9 Predicted average noise for 8 children is 9 db Y is the notation for the predicted Y value this is to distinguish it from Y data that has been measured

21 Movie Example! Number of Tickets Sold (X Variable) Y = bx + a a = Y intercept = 0 Theatre Revenue (Y Variable) Theatre Revenue = 11 Therefore, Y = 11X Number of Tickets Sold

22 Question! Using the formula Y = 11X+0, determine the amount of revenue the theatre would generate on the sale of 250 movie tickets a) 2750 b) 2500 c) 2700 d) 275 X=250, we want Y. Y=11* Y = 2750

23 Question! What would it mean if b = -0.50? a. For every 1 unit increase in X, Y increases by 0.50 b. For every 1 unit increase in X, Y decreases 0.50 units c. X and Y have an inverse (negative) relationship d. X and Y have a direct (positive) relationship e. b and c This one!

24 Perfect vs. imperfect relationship! So far considered perfect relationships, but most relationships in psychology are imperfect not all data falls perfectly on a straight line. Often best that can be done is to draw line of best fit Best fitting line often used for prediction, if so it is called a regression line Shows the imperfect relationship between IQ and GPA Regression line can be used to predict IQ from GPA or vice versa

26 Back to correlation! Plotting these graphs allows us to visualise the relationship between variables. Correlation informs us about the direction and degree of that relationship Measured using correlation coefficients This data varies in the direction and degree in the relationship between the variable this is what correlation coefficients attempt to express

27 Correlation Coefficients! = r values between -1 and +1 which indicate the magnitude and direction of relationship sign +/- indicates the direction of relationship value of the number indicates the magnitude of relationship 1 (or -1) indicates perfect correlation such that knowing one variable allows you to perfectly predict the other 0 indicates zero correlation, knowing one variable does not allow you to predict the the other Imperfect relationships have values between 0 and 1 where knowing one variable helps predict the other with some inaccuracy

28 Correlation co-eficients The closer the data points to the regression line, the more accurate the prediction and the closer the correlation co-efficient to 1 (or -1 if negative) Toward +1 Toward -1 When no relationship between X and Y correlation = 0 Correlation is negative when relationship is negative

29 Question! Which of the following correlation coefficients represents the strongest relationship between two variables? a).2 b).5 c) -.4 d) -.5 e) b and d Both b and d are equally strong correlation co-efficients, strength is expressed in the absolute magnitude of the numbers

30 Question! Based on this graph what is the correlation between ticket sales and theatre revenue? a) -1 b) +1 c) 0 d) 0.5 e) Cannot be determined Ta da! Theatre Revenue Number of Tickets Sold

31 Calculate Correlation Coefficient! Problem: How do we correlate two variables which were measured using different scales and in different units? Example: How can we correlate height in inches and weight in pounds? The variables are on different measurement scales Answer: Z-scores Hey man, by getting the into the same language of z- scores, we can measure relationships between 2 variables independently of scaling and units of measurement. Awesome!

32 Z-Score Method! This equation allows you to calculate correlation coefficient from z-scores Sum of the product of each z-score pair r = z X z Y n 1 Pearson s r is the measure of correlation we are calculating Number of z-score pairs in sample, with minus 1 as a trick to compensate for sample having lower variance note: the book uses large N

33 Z-Score Method! r = z X z Y n 1 Use this to calculate correlation between OJ consumption and doctor s visits Subject Glasses of OJ Per Day (X) if we have the means and standard deviations we can calculate the z-scores = 1.5 = 5.5 S x = S Y = Doctor s Visits Per Year (Y)

34 Z-Score Method step 1! Convert all of the Scores to z-scores Subject OJ (X) Doctor (Y) z x z y

35 Z-Score Method - step 2 r = z X z Y n 1 Calculate the products of the z-scores (i.e., multiply them together) Subject OJ (X) Doctor (Y) z x z y z x z y zxzy = Then sum the product of the z- scores

36 Z-score method step 3! Solve the Equation This method can take a long time to convert each score into a z- score, and can cause error through rounding. What can you conclude from this? The correlation between glasses of OJ drank per day and doctor s visits per year is r = -0.94

37 Correlation Coefficient Shortcut! Believe it or not this beast is a shortcut! r = X 2 XY ( X) 2 n ( X) Y n Y 2 ( ) ( Y) 2 n By rearranging the equation can be transformed so that you can input raw scores instead of z- scores Although it looks complicated each element of the equation can be simply calculated in SPSS

38 Correlation Coefficient Shortcut! r = X 2 XY ( X) 2 n ( X) Y n Y 2 ( ) ( Y) 2 n Just create a column for each element Subject X Y X 2 Y 2 XY Sum them =9 = 33 = 27 = 207 = 32

39 Correlation Coefficients Shortcut! Calculate value of each element from previous spreadsheet = 9 r = X 2 XY ( X) 2 n ( X) Y n Y 2 ( ) ( Y) 2 n Plug values into equation = 33 = 27 = 207 = 32 r = [ ][ ] n = 6 r = -.94

40 Another interpretation of r! r = Proportion of total Variability of Y that is accounted for by X You can think of variability accounted for by X as being determined by how close the data is to the line of regression (line of best fit) When all of the variability in Y is accounted for by X then r = 1 When only some of the variability in Y is accounted for by X then r < 1 When none of the variability in Y is accounted for by X then r = 0

41 r 2 = coefficient of determination! r is a square root of something, which is hard to conceptualise r = Proportion of total Variability of Y that is accounted for by X r 2 = Proportion of total variability of Y that is accounted for by X By squaring r we get a more intuitively meaningful quantity

42 Variability of Y Accounted by X! Another way to try to understand same concept in terms of r 2 Total variability in Y (doctor visits / yr) Total variability in X (Glasses of OJ / day ) variability in Y accounted for by X (Variability in doctors visits accounted for by OJ consumption)

43 Variability of Y Accounted by X! r 2 = The proportion of the total variability of Y that is accounted for by X r 2 is high High proportion of variability of doctors visits (Y) can be accounted for by variability in OJ consumption (X) r 2 = purple (variability of Y accounted for by X) as a proportion of blue (total variability in Y) r 2 is low Low proportion of variability of doctors visits (Y) can be accounted for by variability in OJ consumption (X)

44 Coefficient of Determination r 2! To convert between r and r 2 you simply square the r value If r = -.94 r 2 = Therefore 88.35% of the variability of doctor s visits is accounted for by orange juice r can give the impression that a factor is more influential than it really, really you should be looking at r 2 which tells you how much of the change in Y can be accounted for by X, which is always lower (except if r = 1 in which case they are identical) When measuring behaviour r = 0.5 is pretty high, yet in r 2 this accounts for only 25% of variability in behaviour

45 Question! If r = 0.6 how much of the variability of Y is accounted for (explained) by X a) 6% b) 60% c) 36% d) 3.6% e) 66% 0.6 squared then converted to % is 36%

46 Other correlation coefficients! There are many different correlation coefficients Pearson s r is most common for analysing behaviour, but there are others Which is best depends on the shape of the relationship and the measuring scale used e.g. linear vs. curvilinear e.g. interval, ratio, ordinal

47 Varieties of Correlation Coefficients! Pearson Correlation Coefficient (r) Calculated when the relationship is linear and both variables are measured on an interval or ratio scale (e.g., IQ and weight) Spearman Rank Order Correlation Coefficient (r s ) Calculated when the relationship is linear and at least one of the variables are measured on an ordinal scale (e.g., ranking in a competition) Biserial Correlation Coefficient (r b ) Calculated when the relationship is linear and one variable is measured on an interval or ratio scale and the other is dichotomous (e.g., IQ and gender) Phi Coefficient (φ) Calculated when the relationship is linear and both variables are dichotomous (e.g., Gender and handedness) Eta(η) Calculated when the relationship between the variables is curvilinear (e.g., anxiety and test performance) IQ commonly treated as interval even though it is not really interval in exam / assignments you can treat it as interval This means is in either one state or another male or female, dead or alive, human or nonhuman

48 Spearman Rank Order (r s )! Judge A and Judge B both rank order 9 students according to how extraverted they behave If we want to know whether there is reliability between the judges we can calculate the correlation between their rankings (inter-rater reliability) Since both variables are ordinal we will use spearman s rank order This is equivalent to Pearson s r but for ordinal data that is assumed to be linear

49 Spearman Rank Order equation! r s =1 6 D i n 3 n 2 Where, Di = difference between ith pair of ranks, n = number of pairs of ranks Describes the linear relationship between two variables on ordinal scales For the love of god, do I have to spell everything that is self-evident out for you, this is merely a simplification of. r = z X z Y n 1

50 Calculating Rho (r s )! If judge B couldn t decide between Debbie and Charlotte and tied them (gave them both 6) then their ranks would both be changed to 5.5 Contestant Judge A s Ranking Judge B s Ranking Difference in ranking D D 2 Cindy Paige Maria Debbie CharloGe Brook Jennifer Monique Shannon Difference in ranking squared = 18 Plug numbers in r s =1 6 D i n 3 n 2 n = 9

51 Effect of range restriction! If a correlation exists between X and Y, restricting the range has the effect of lowering the correlation Stress level If look at the restricted range between here, then the data looks more like a cloud of data with less of a systematic relationship between X and Y, this is reflected in the calculated correlation co-efficient Sometimes low correlations can be measured due to overly restrictive ranges salary

53 When scientists get annoyed! Because correlation is so widespread it is also inevitably widely misinterpreted It is very common for people to assume that if two things are correlated then one must have caused the other

54 Correlation Causation! There are 3 ways in which correlation can occur in the absence of causation: 1. Spurious correlation - the correlation is unique to the to the sample by chance 2. Directionality problem - cannot determine if X causes Y or Y causes X 3. Third variable problem variability in a third unobserved variable is responsible for the correlation In this case the correlation will disappear when study replicated This is why it is so important to replicate studies This is a problem with observational studies, since you are not intervening on reality and changing it you cannot say what is causing what in which direction

55 Angry hamsters! Correlation between hamsters getting angry and laboratory ice melting is highly correlated in the laboratory of paranormal pseudoscience Does ice melting cause hamsters to be angry? Vice versa? Could be due to third variable of temperature which causes both.

56 Assignment 2! The assignment involves taking the variables from the group survey on survey monkey, and probing the data for correlations between the different variables, using some of the techniques learnt today. Today's demo will demonstrate some of the functions in SPSS you will need to complete the assignment.

57 Next Lecture! Regression! Hell yeah!