Can you tell the relationship between students SAT scores and their college grades?

Transcription

1 Correlation

2 One Challenge Can you tell the relationship between students SAT scores and their college grades? A: The higher SAT scores are, the better GPA may be. B: The higher SAT scores are, the lower GPA may be. C: SAT and GPA are not related at all How do you make your choice? Common sense How sure are you about your answer?

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4 Correlation A statistics method to measure the relationship between two variables Three characteristics Direction of the relationship Form of the relationship Strength/Consistency

5 Direction of the Relationship Positive correlation Variables moving in the same direction Negative correlation Variables moving in opposite directions

6 Form of the Relationship Linear or non-linear Predicting data

7 Strength/Consistency How well do data fit the specific form? Measured by the distance between actual data and the predicted data The absolute value of a correlation Measuring the fitness 1: Perfect fit 0: Not fit at all

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9 Correlation Measures The Pearson correlation Linear relationship The sign of the correlation: direction The numerical value: the degree of the relationship The Spearman correlation For ordinal scale of measurement Both the X values and the Y values are ranks. Measuring consistency for data relationship Not necessarily to be linear The point-biserial correlation Used to measure the correlation between a regular variable and a dichotomous variable

10 The Pearson Correlation degree to which X and Y vary together r degree to which X and Y vary separately covariability of X and Y variability of X and Y separately SP SS X SS Y

11 SP SS ( X M X )( Y MY ) ( X M ) 2 SS ( X ) 2 X n SP XY 2 ( X )( Y) n

12 Example X Y XY S SX 2 =244 SY 2 =24

13 Check the Result Using the scatterplot of data Drawing the envelope around all data points Checking the direction and shape of the envelope

14 Interpreting Correlations Predication Correlation is just about relationship between two variables. Not necessarily causal!!! The value could be affected greatly by the data range. Outliers can dramatically affect the value.

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16 Data Range and Correlation

17 Outlier and Correlation

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19 The Strength of Relationship The coefficient of determination Squaring the value of correlation How much of the variance in dependent variable is accounted for by independent variable. Similar to the power used in z- and t-tests

20 Hypothesis Tests with the Pearson Correlation Pearson correlation is usually computed for sample data, but used to test hypotheses about the relationship in the population Population correlation shown by Greek letter rho (ρ) Non-directional: H 0 : ρ = 0 and H 1 : ρ 0 Directional: H 0 : ρ 0 and H 1 : ρ > 0 or Directional: H 0 : ρ 0 and H 1 : ρ < 0

21 Population vs. Sample

22 df What should the df be?

23 Correlation Hypothesis Test Sample correlation r used to test population ρ Degrees of freedom (df) = n 2 Hypothesis test can be computed using either t or F Use t table to find critical value with df = n - 2 t r 2 (1 r ) ( n 2)

24 Example 14.6 & 14.7 α =.05 n = 30 r = 0.35 Two-tailed test: critical value ±2.048 Fail to reject the null hypothesis One-tailed test: reject: Reject

25 Report Correlations A correlation for the data revealed a significant relationship between amount of education and annual income, r = +0.65, n = 30, p <.01, two tails.

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28 Partial Correlation A partial correlation measures the relationship between two variables while mathematically controlling the influence of a third variable by holding it constant r xy z r xy (1 r 2 xz ( r xy r yz )(1 ) r 2 yz )

29 Example 14.8 Number of Churches (X) Number of Crimes (Y) Population (Z)

30 What if the relationship looks like this?

31 The Spearman Correlation To measure the degree of consistency of direction Not necessarily linear. One extra step before calculating the Pearson correlation Ranking the X and Y values Analyze the correlation of ranking values. X Y (values) X Y (Ranks)

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33 Ranking Tied Scores Using the same rank for same scores Ranking all scores Computing the mean for ranked position of same scores X Y (values) X Y (Ranks) (2.5) (2.5)

34 Special Formula for Spearman Correlation SS n( n 2 1) 12 r s n( n D 2 1)

35 The Point-Biserial Correlation Just like the Pearson correlation One variable has only two values Gender, success/failure, college education or not, The value of correlation has nothing to do with the values you use

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37 Correlation in Excel

38 If we know two variables are linearly related, how can we describe such a relationship? Using a linear equation y = bx + a

39 Regression

40 Goal of Regression Determining two constants for a linear equation: y=bx+a b: slope a: intercept Methods The least-squares solution

41 ^ Distance = Y - Y ^ Minimizing S(Y-Y) 2

42 Formula b SP SS X a M Y bm X

43 Example X Y XY S SX 2 =244 SY 2 =24

44 Regression in Excel Draw a scatterplot Show the trendline

45 Linear Equations and Regression The Pearson correlation measures a linear relationship between two variables This figure Makes the relationship easier to see Shows the central tendency of the relationship Can be used for prediction

46 Linear Equations General equation for a line Equation: Y = bx + a X and Y are variables a and b are fixed constant

47 Regression Regression is a method of finding an equation describing the best-fitting line for a set of data Least square Minimizing errors of known data Or the error of prediction

48 Error of Prediction With a linear function from regression, we can calculate the predicted value based on a given X Ŷ Error of prediction: Y- Ŷ Often squared

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50 Standard Error of Estimate Regression equation makes a prediction Precision of the estimate is measured by the standard error of estimate (SEoE) SEoE = SS residual df ( Y Yˆ) n 2 2 SS residual ( ˆ) Y Y n 2 2

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52 Relationship Between Correlation and Standard Error of Estimate SS regression = r 2 SS Y SS residual = (1 - r 2 ) SS Y SS df residual 2 (1 r ) SS n 2 Y

53 Testing Regression Significance Analysis of Regression Similar to Analysis of Variance Uses an F-ratio of two Mean Square values Each MS is an SS divided by its df H 0 : the slope of the regression line (b or beta) is zero no regression

54 Mean Squares and F-ratio MS regression SS df regression regression MS residual SS df residual residual F MS MS regression residual

55 SS and df in Regression Analysis

56 SPSS Output Example

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58 In Summary Correlation: the relationship between two variables Direction, form, degree Three methods For different purposes Regression Determining the linear equation that data best fit Slope and intercept

59 Homework Chapter 14: 10, 26, 28

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63 Next Week No class on Tuesday Thursday Chi-square test