Hypothesis Test Hypotheses. A. H 0 : μ 1 = μ 2 against H a : μ 1 ≠ μ 2 (two-sided) [ H a : μ 1 > μ 2 (right-sided) H a : μ 1 < μ 2 (left-sided) ] Test statistic. B. 2 2 ( x x ) s s 1 2 1 2 t where SE stat x x 1 2 SE n n x x 1 2 1 2 df Welch C. P -value. Convert the t stat to P- value with t table or software. Interpret. D. Significance level (optional). Compare P to prior specified α level. 35
Hypothesis Test – Example A. Hypotheses. H 0 : μ 1 = μ 2 vs. H a : μ 1 ≠ μ 2 B. Test stat. In prior analyses we calculated sample mean difference = 34.75 mg/dL, SE = 13.563 and df conserv = 19. ( x x ) 34.75 1 2 t 2.56 with 19 df stat SE 13.563 x x 1 2 C. P -value. P = 0.019 → good evidence against H 0 (“significant difference”). D. Significance level ( optional ). The evidence against H 0 is significant at α = 0.05 but not at α = 0.01. 36
Equal Variance t Procedure Also called pooled variance t procedure Not as robust as prior method, but… Historically important Calculated by software programs Leads to advanced ANOVA techniques 37
Pooled variance procedure We start by calculating this pooled estimate of variance 2 2 ( df )( s ) ( df )( s ) 2 1 1 2 2 s pooled df df 1 2 where 2 is the variance in group and s i i 1 df n i i 38
The pooled variance is used to calculate this standard error estimate: 1 1 2 SE s x x pooled 1 2 n n 1 2 Confidence Interval ( x x ) ( t )( SE ) 1 2 x x df , 1 1 2 2 Test statistic ( x x ) 1 2 t stat SE x x 1 2 All with df = df 1 + df 2 = ( n 1 − 1) + ( n 2 − 1) 39
Pooled Variance t Confidence Interval Group n i s i xbar i Data 1 20 36.64 245.05 2 20 48.34 210.30 1 1 SE 1839.623 13 . 56 x x 1 2 20 20 df ( 20 1 ) ( 20 1 ) 38 95 % CI for ( x x ) ( t )( SE ) x 1 2 1 2 38 ,. 975 x 1 2 ( 245 . 05 210 . 30 ) ( 2 . 02 )( 13.56 ) 34 . 75 27 . 39 (7.36, 62.14) 40
Pooled Variance t Test Data: Group xbar i n i s i 1 20 36.64 245.05 2 20 48.34 210.30 1 1 SE 1839.623 13 . 56 x x 1 2 20 20 df ( 20 1 ) ( 20 1 ) 38 H : H : 0 1 2 a 1 2 34 . 75 x x 1 2 t 2.56; df 38 stat SE 13 . 56 x x 1 2 41 P 0 . 014
How to do it in Excel? Data set: Presentation3_FCL.xlsx First do Levene’s test to see whether two group has equal variance 42
How to do it in Excel? Levene’s test shows no significant difference in variance between groups 43
How to do it in Excel? Next use t-Test: Two-Sample Assuming Equal Variances for the test 44
How to do it in Excel? Click OK to get results 45
Excel results p-value = 0.014 Significant difference in fasting cholesterol levels between Type A personality subjects and Type B personality subjects. 46
How to do it in JMP? Data set: Presentation3_FCL.jmp Analyze --- Fit Y by X 47
How to do it in JMP? Select Means/ANOVA/ Pooled t for the equal variance t test Select t Test for unequal variance t test 48
Results from JMP p-value <0.05 Significant difference in fasting cholesterol levels between Type A personality subjects and Type B personality subjects. 49
Conditions for Inference Conditions required for t procedures: “Validity conditions” a. Good information (no information bias) b. Good sample (“no selection bias”) c. “No confounding” “Sampling conditions” a. Independence b. Normal sampling distribution 50
ANOVA 51
Illustrative Example: Data Pets as moderators of a stress response . This chapter follows the analysis of data from a study in which heart rates (bpm) of participants were monitored after being exposed to a psychological stressor. Participants were randomized to one of three groups: Group 1 - monitored in presence of pet dog Group 2 - monitored in the presence of human friend Group 3 - monitored with neither dog nor human friend present 52
Illustrative Example: Data 53
Descriptive Statistics Data are described and explored before moving to inferential calculations Here are summary statistics by group: 54
Side-by-Side Boxplots 55
Analysis of Variance One-way ANalysis Of VAriance (ANOVA) Categorical explanatory variable Quantitative response variable Test group means for a significant difference Statistical hypotheses H 0 : μ 1 = μ 2 = … = μ k H a : at least one of the μ i s differ Method: compare variability between groups to variability within groups ( F statistic) 56
Analysis of Variance, cont. R. A. Fisher (1890-1962) The F in the F statistic stands for “Fisher” 57
Mean Square Between: Graphically 58
Mean Square Between: E xample 59
Mean Square Within: Graphically 60
Mean Square Within: Example 61
The e F F sta stati tisti stic a c and nd AN ANOVA A ta table Data are arranged to form an ANOVA table F statistic is the ratio of the MSB to MSW F stat “signal -to- MSB 1193 . 843 F stat 14 . 08 noise” ratio MSW 84 . 793 62
F stat and P -value The F stat has numerator and denominator degrees of freedom: df 1 and df 2 respectively (corresponding to df B and df W ) Convert F stat to P -value with a computer program The P -value corresponds to the area in the right tail beyond 63
F stat and P -value P < 0.001 64
How to do one-way ANOVA in EXCEL? Data set: Presentation3_pet.xlsx 65
How to do one-way ANOVA in EXCEL? 66
Analysis results from Excel One-way ANOVA shows significant difference in mean FEV values among the four different smoker groups. 67
How to do one-way ANOVA in JMP? Presentation3_pet.jmp file Analyze---Fit Y by X 68
How to do one-way ANOVA in JMP? 69
Analysis results from JMP Pairwise comparisons from Tukey’s method shows the signficant difference among all three groups. 70
Correlation and simple linear regression 71
Data type for correlation and regression Quantitative response variable Y (“dependent variable”) Quantitative explanatory variable X (“independent variable”) Historically important public health data set used to illustrate techniques (Doll, 1955) n = 11 countries Explanatory variable = per capita cigarette consumption in 1930 (CIG1930) Response variable = lung cancer mortality per 100,000 (LUNGCA) 72
Data, cont. 73
Scatterplot Bivariate ( x i , y i ) points plotted as scatter plot. 74
Inspect scatterplot’s Form: Can the relation be described with a straight or some other type of line? Direction : Do points tend trend upward or downward? Strength of association: Do point adhere closely to an imaginary trend line? Outliers (in any): Are there any striking deviations from the overall pattern? 75
Judging Correlational Strength Correlational strength refers to the degree to which points adhere to a trend line The eye is not a good judge of strength. The top plot appears to show a weaker correlation than the bottom plot. However, these are plots of the same data sets. (The perception of a difference is an artifact of axes scaling.) 76
Correlation Correlation coefficient r quantifies linear relationship with a number between −1 and 1. When all points fall on a line with an upward slope, r = 1. When all data points fall on a line with a downward slope, r = −1 When data points trend upward, r is positive; when data points trend downward, r is negative. The closer r is to 1 or −1, the stronger the correlation. 77
Examples of correlations 78
Calculating r (Pearson Correlation) Formula Correlation coefficient tracks the degree to which X and Y “go together.” Recall that z scores quantify the amount a value lies above or below its mean in standard deviations units. When z scores for X and Y track in the same direction, their products are positive and r is positive (and vice versa). 79
Calculating r, Example 80
Scatter plot and r in Excel Data set: Presentation3_CIGLungCA.xlsx Scatter plot: select the data in column B and C---Insert --- Scatter Plot---Add x and y axis labels Correlation: Lung cancer mortality vs. Data analysis---correlation cigarette consumption 50 45 40 Lung cancer mortality 35 30 25 20 15 10 5 0 0 200 400 600 800 1000 1200 1400 CIG1930 81
Scatter plot and r in JMP Data set: Presentation3_CIGLungCA.jmp Scatter plot: Graph---Scatter plot matrix 82
Interpretation of r Direction. The sign of r indicates the direction of the 1. association: positive ( r > 0), negative ( r < 0), or no association ( r ≈ 0). Strength. The closer r is to 1 or −1, the stronger the 2. association. Coefficient of determination. The square of the 3. correlation coefficient ( r 2 ) is called the coefficient of determination. This statistic quantifies the proportion of the variance in Y [mathematically] “explained” by X. For the illustrative data, r = 0.737 and r 2 = 0.54. Therefore, 54% of the variance in Y is explained by X. 83
Notes, cont. 4. Reversible relationship. With correlation, it does not matter whether variable X or Y is specified as the explanatory variable; calculations come out the same either way. [This will not be true for regression.] 5. Outliers. Outliers can have a profound effect on r . This figure has an r of 0.82 that is fully accounted for by the single outlier. 84
Notes, cont. 6. Linear relations only. Correlation applies only to linear relationships This figure shows a strong non-linear relationship, yet r = 0.00. 7. Correlation does not necessarily mean causation. Beware lurking variables (next slide). 85
Confounded Correlation A near perfect negative correlation ( r = −.987) was seen between cholera mortality and elevation above sea level during a 19th century epidemic. We now know that cholera is transmitted by water. The observed relationship between cholera and elevation was confounded by the lurking variable proximity to polluted water. 86
Hypothesis Test Random selection from a random scatter can result in an apparent correlation We conduct the hypothesis test to guard against identifying too many random correlations. 87
Hypothesis Test A. Hypotheses. Let ρ represent the population correlation coefficient. H 0 : ρ = 0 vs. H a : ρ ≠ 0 (two -sided) [or H a : ρ > 0 (right-sided) or H a : ρ < 0 (left-sided)] B. Test statistic 2 r 1 r t where SE stat r SE n 2 r df n 2 C. P -value. Convert t stat to P -value with software or t table. 88
Hypothesis Test – Illustrative Example H 0 : ρ = 0 vs. H a : ρ ≠ 0 (two -sided) A. B. Test stat 2 1 0 . 737 0.2253 SE r 11 2 0 . 737 t 3.27 stat 0.2253 df 11 2 9 C. .005 < P < .01 by Table C. P = .0097 by computer. The evidence against H 0 is highly significant. 89
Exercise (True/False) 1. Correlation coefficient r quantifies the relationship between quantitative variables X and Y . 2. The closer r is to 1, the stronger the linear relation between X and Y . 3. If r is close to zero, X and Y are unrelated. 4. The value of r changes when the units of measure are changed. 90
Regression Regression describes the relationship in the data with a line that predicts the average change in Y per unit X. The best fitting line is found by minimizing the sum of squared residuals, as shown in this figure. 91
Regression Line, cont. The regression line equation is: where ŷ ≡ predicted value of Y , a ≡ the intercept of the line, and b ≡ the slope of the line Equations to calculate a and b SLOPE: INTERCEPT: 92
Regression Line, cont. Slope b is the key statistic produced by the regression 93
Calculate regression line in Excel Data analysis---regression 94
Calculate regression line in JMP Analyze---fit Y by X 95
Conditions for Inference Inference about the regression line requires these conditions Linearity Independent observations Normality at each level of X Equal variance at each level of X 96
Conditions for Inference This figure illustrates Normal and equal variation around the regression line at all levels of X 97
Assessing Conditions The scatterplot should be visually inspected for linearity, Normality, and equal variance Plotting the residuals from the model can be helpful in this regard. The table lists residuals for the illustrative data 98
Assessing Conditions, cont. A stemplot of the residuals show |-1|6 no major departures from |-0|2336 Normality | 0|01366 | 1|4 x10 This residual plot shows more variability at higher X values (but the data is very sparse) 99
Residual Plots With a little experience, you can get good at reading residual plots. Here’s an example of linearity with equal variance. 100
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