Correlation Chapter 11 PDF

Correlation Chapter 11 1 What have we learned so far? ➻Describe a distribution ➻Measures of central tendency ➻Mode ➻Median ➻Mean ➻Measures of variability ➻Range ➻Variance ➻Standard Deviation 2  What is Research? ➻Trying to unde...

Correlation Chapter 11 1 What have we learned so far? ➻Describe a distribution ➻Measures of central tendency ➻Mode ➻Median ➻Mean ➻Measures of variability ➻Range ➻Variance ➻Standard Deviation 2  What is Research? ➻Trying to understand the world better ➻Looking for similarities or differences ➻We have done this for ages! ➻When it snows, animals disappear ➻Early humans might have gone hungry! ➻How do we formalize this concept? ➻Relating two types of measurements ➻Examples in research? 3  Relating Variables ➻Basis of scientific research ➻Relationship among a set of variables ➻Measure of similarity ➻Knowing about one tells us about another ➻Height and weight ➻Tall people are generally heavier ➻Height is partially redundant with weight ➻Measures of co-relating variables  Correlation ➻Relationship between two variables ➻Specifically, a linear relationship ➻Let’s use an example – and count! ➻In six towns: # of ice average # Town cream of fires per ➻Number of ice stands month cream stands A 1 1 B 2 1 ➻Average number C 2 2 of fires per month D 3 2 ➻Is there a pattern? E 4 3 F 5 4 5  Drawing a Scatter Plot ➻Plot the numbers on X axis and Y axis ➻All rules of plotting still apply Scatter Plot for Number of Ice Cream Stands & Number of Fires per Month # of ice average # 6 Number of Fires Per Month Town cream of fires stands per month 5 A 1 1 4 B 2 1 3 C 2 2 2 D 3 2 1 E 4 3 0 F 5 4 0 1 2 3 4 5 6 Number of Ice Cream Stands 6  Interpreting a Scatter Plot ➻ As measurement on one variable increases ➻ Measurement on the other variable increases ➻ What does this mean? ➻ Ice cream stands cause fires? ➻ Fires are hot, need ice cream to cool down? ➻ Simply tells you there is a relationship ➻ Does NOT tell you there is a causation ➻ Correlation DOES NOT IMPLY causation ➻ Repeat after me! 7  More Examples ➻How many times do people brush their teeth in a day? ➻How many cavities do they have? # of times they # of cavities they Patient brush per day have A 1 5 B 2 4 C 3 4 D 4 3 E 6 2 F 7 1 8  Drawing a Scatter Plot ➻Plot number on X axis and Y axis ➻Choose your axes carefully Scatter Plot for Number of Times Patients Brush Teeth Per Day and Number of Cavities # of times # of 6 Patient they brush cavities per day they have 5 Number of Cavities A 1 5 4 B 2 4 3 C 3 4 2 D 4 3 1 E 6 2 0 0 1 2 3 4 5 6 7 F 7 1 Number of Times Patients Brush Teeth Per Day 9  More Examples ➻Number of servings of vegetables ➻Grade on statistics test Servings of Grade on statistics Student vegetables per year test A 20 20 B 20 80 C 60 60 D 100 20 E 100 100 10  Drawing a Scatter Plot ➻Plot number on X axis and Y axis ➻Choose your axes carefully Scatter Plot for Number of Servings of Vegetables Per Year & Grade on Statistics Test Servings of Grade on 120 Student vegetables statistics Geade on Statistics Test per year test 100 80 A 20 20 60 B 20 80 40 C 60 60 20 D 100 20 0 0 20 40 60 80 100 120 E 100 100 Number of Servings of Vegetables Per Year 11  Interpreting Relationship ➻Direction of relationship ➻Look at the pattern of the data points ➻Positive relationship ➻As one goes UP, other goes UP ➻Negative relationship ➻As one goes UP, other goes DOWN ➻Zero or no relationship ➻There is no linear pattern (i.e., not a line) 12 Name the Direction of Relationship Negative Positive Relationship Relationship No Curvilinear Relationship Relationship 13  Interpreting Relationship ➻Magnitude of relationship ➻Assigning a number ➻Should it depend on number of points? ➻Coefficient of Correlation (r) ➻Always between –1 and + 1 Perfect Perfect Negative No Positive Relationship Relationship Relationship -1 -0.5 0 +0.5 +1 14 Interpreting the Correlation Coefficient ➻Sign of r: direction of correlation ➻– signifies a negative correlation ➻+ signifies a positive correlation ➻Value of r: strength of the correlation ➻Higher the value, greater the magnitude ➻Lower the value, lesser the magnitude ➻Which correlation is stronger? ➻0.25 or –0.75? 15  Formalizing these Concepts ➻Pearson Coefficient of Correlation (r) The Numerator The Denominator Sum of Deviation Products Sum of Squares (SDP) (For Each Variable) 16  Let’s Break that Down Positive Correlation Negative Correlation ➻ Most high X values paired ➻ Most high X values paired with high Y values with low Y values ➻ Most low X values paired ➻ Most low X values paired with low Y values with high Y values 17  Let’s Break that Down ➻Sum of Deviation Products (SDP) ➻Deviation from respective means ➻A value above the mean ➻X – MX = + number ➻Y – MY = + number ➻A value below the mean ➻X – MX = – number ➻Y – MY = – number 18  Let’s Break that Down ➻ Positive Correlation: ➻ Negative Correlation: ➻ No Correlation: 19 Pearson Correlation Coefficient ➻Find the association between attendance and class performance Attendance Performance Scatter Plot for Attendance & Student (X) (Y) Performance 4 A 1 1 3 Performance 2 B 2 2 1 C 4 2 0 0 1 2 3 4 5 6 D 5 3 Attendance 20 Computing the Correlation Coefficient X MX (X – MX) (X – MX)2 Y MY Y – MY (Y – MY)2 (X – MX) (Y – MY) 1 3 –2 4 1 2 –1 1 (–2) × (–1) = 2 2 3 –1 1 2 2 0 0 (–1) × (0) = 0 4 3 +1 1 2 2 0 0 (+1) × (0) = 0 5 3 +2 4 3 2 +1 1 (+2) × (+1) = 2 SSX = 10 SSY = 2 SDP = 4 Sum of Deviation Products (SDP) 21 Computing the Correlation Coefficient Positive Correlation 22 Pearson Correlation Coefficient ➻Find the association between late- night parties and class performance Parties Performance Scatter Plot for Parties and Performance Student (X) (Y) 4 A 1 3 3 Performance 2 B 2 2 1 C 4 2 0 0 1 2 3 4 5 6 D 5 1 Parties 23 Computing the Correlation Coefficient X MX (X – MX) (X – MX)2 Y MY Y – MY (Y – MY)2 (X – MX) (Y – MY) 1 3 –2 4 3 2 +1 1 (–2) × (+1) = –2 2 3 –1 1 2 2 0 0 (–1) × (0) = 0 4 3 +1 1 2 2 0 0 (+1) × (0) = 0 5 3 +2 4 1 2 –1 1 (+2) × (–1) = –2 SSX = 10 SSY = 2 SDP = – 4 Sum of Deviation Products (SDP) 24 Computing the Correlation Coefficient – – – – Negative Correlation 25  Squared Coefficient of Correlation ➻ Measures strength of association of variables ➻ Proportion of variance accounted for ➻ Proportion of variance in common (or shared) ➻ Coefficient of determination ➻ Proportionate reduction of error ➻ If r = 0.89, then r2 = (0.89)2 = 0.79 ➻ The two variables share 79% of the total variance X 79% Y 26  Outlying Points ➻Influence correlation coefficient 27  Some Issues with r ➻ Only indicates linear relationship ➻ Can’t use for curvilinear relationships ➻ Sampling problems ➻ Truncated or restricted range ➻ Weak internal validity ➻ Can’t determine cause and effect 28  Correlation ≠ Causality ➻ Three possible directions for causality 1. X could be causing Y 2. Y could be causing X 3. A third variable Z could be causing both X and Y 29  Studying Relationship ➻Direction and magnitude of relation ➻Pearson coefficient of correlation (r) ➻Values between –1 and + 1 ➻Computing coefficient of correlation ➻Squared coefficient of correlation ➻Issues with coefficient of correlation ➻Correlation DOES NOT IMPLY causation 30