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Vooruitblik 10 en 11 Dinsdag 30 september 2008. Chapter 10 Correlation and Regression 1. Correlation 2. Regression 3. Variation and Prediction Intervals.

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Presentation on theme: "Vooruitblik 10 en 11 Dinsdag 30 september 2008. Chapter 10 Correlation and Regression 1. Correlation 2. Regression 3. Variation and Prediction Intervals."— Presentation transcript:

1 Vooruitblik 10 en 11 Dinsdag 30 september 2008

2 Chapter 10 Correlation and Regression 1. Correlation 2. Regression 3. Variation and Prediction Intervals 4. Rangorde correlatie

3 1. Correlation Verband tussen twee gemeten variabelen in een dataset op interval of ratio nivo In dit boek: alléén lineaire verbanden Let op de voorwaarden! Maat: Pearson PM correlatie r of rho Geen correlatie: r = 0, maximale correlatie r = -1 of +1 Kritische waarden: tabel A-6

4 Scatterplots of Paired Data Figure 10-2

5 Scatterplots of Paired Data Figure 10-2

6 Formula 10-1 n  xy – (  x)(  y) n(  x 2 ) – (  x) 2 n(  y 2 ) – (  y) 2 r =r = The linear correlation coefficient r measures the strength of a linear relationship between the paired values in a sample. Calculators can compute r Formula

7 Figure 10-3 Hypothesis Test for a Linear Correlation

8 2. Regression Vervolg op correlatie Berekening van regressielijn in de scatterplot: de lijn die het beste past in de puntenwolk Doel: voorspellen van waarden

9 Regression The typical equation of a straight line y = mx + b is expressed in the form y = b 0 + b 1 x, where b 0 is the y -intercept and b 1 is the slope. ^ The regression equation expresses a relationship between x (called the independent variable, predictor variable or explanatory variable), and y (called the dependent variable or response variable). ^

10 Formulas for b 0 and b 1 Formula 10-2 n(  xy) – (  x) (  y) b 1 = (slope) n(  x 2 ) – (  x) 2 b 0 = y – b 1 x ( y -intercept) Formula 10-3 calculators or computers can compute these values

11 Given the sample data in Table 10-1, find the regression equation. Example: Old Faithful - cont

12 Procedure for Predicting Figure 10-7

13 3. Variation and Prediction Intervals Vervolg op regressielijn (hfst 7) Confidence interval = interval schatting van populatie parameters: proportie, gemiddelde, variantie Hier: interval schatting van de schatting van de waarde van een variabele

14 Key Concept In this section we proceed to consider a method for constructing a prediction interval, which is an interval estimate of a predicted value of y.

15 y - E < y < y + E ^ ^ Prediction Interval for an Individual y where E = t   2 s e n(x2)n(x2) – (  x) 2 n(x0 – x)2n(x0 – x) n x 0 represents the given value of x t   2 has n – 2 degrees of freedom

16 Standard Error of Estimate The standard error of estimate, denoted by s e is a measure of the differences (or distances) between the observed sample y -values and the predicted values y that are obtained using the regression equation. Definition ^

17 4. Rangorde correlatie Non-parametrische methode = verdelingsvrije toets = geen aannames mbt. Verdeling in de opulatie Associatietest op twee variabelen Spearman’s: r s (sample) of voor populatie: rho s Procedure in fig (p.537)

18 voorbeeld

19 1. Goodness-of-fit: multinominaal 2. Kruistabellen (contingency tables) 3. Variantie analyse (ANOVA) Chapter 11 Multinomial Experiments and Contingency Tables

20 Overview  We focus on analysis of categorical (qualitative or attribute) data that can be separated into different categories (often called cells).  Use the  2 (chi-square) test statistic (Table A- 4).  The goodness-of-fit test uses a one-way frequency table (single row or column).  The contingency table uses a two-way frequency table (two or more rows and columns).

21 1. Goodness-of-fit: multinominaal Komt een feitelijke kansverdeling op een nominale variabele overeen met een verwachte verdeling? H0: p1 = x, p2 = y, p3 = z, p4 = etc.. H1: Tenminste één van de gevonden proporties is afwijkend van de verwachte kans.

22 Goodness-of-Fit Test in Multinomial Experiments Critical Values 1. Found in Table A- 4 using k – 1 degrees of freedom, where k = number of categories. 2. Goodness-of-fit hypothesis tests are always right-tailed.  2 =  ( O – E ) 2 E Test Statistics

23 Example: Last Digit Analysis Test the claim that the digits in Table 11-2 do not occur with the same frequency.

24 Relationships Among the  2 Test Statistic, P-Value, and Goodness-of-Fit Figure 11-3

25 2. Kruistabellen (contingency tables) In this section we consider contingency tables (or two-way frequency tables), which include frequency counts for categorical data arranged in a table with a least two rows and at least two columns. We present a method for testing the claim that the row and column variables are independent of each other. We will use the same method for a test of homogeneity, whereby we test the claim that different populations have the same proportion of some characteristics.

26 BlackWhiteYellow/Orange Row Totals Controls (not injured) Cases (injured or killed) Column Totals For the upper left hand cell: = E = (899)(704) 1232 Case-Control Study of Motorcycle Drivers (row total) (column total) E = (grand total)

27 BlackWhiteYellow/Orange Row Totals Cases (injured or killed) Expected Column Totals Controls (not injured) Expected Case-Control Study of Motorcycle Drivers

28 H 0 : Row and column variables are independent. H 1 : Row and column variables are dependent. The test statistic is  2 =  = 0.05 The number of degrees of freedom are (r–1)(c–1) = (2–1)(3–1) = 2. The critical value (from Table A-4) is  2.05,2 = Case-Control Study of Motorcycle Drivers

29 We reject the null hypothesis. It appears there is an association between helmet color and motorcycle safety. Case-Control Study of Motorcycle Drivers Figure 11-4

30 3. Variantie analyse (ANOVA) ANalysis Of VAriance H0 = meerdere populatie gemiddeldes zijn gelijk F-verdeling (tabel A7) Toets op P-waarde

31

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33 TOT SLOT: Bayesiaanse statistiek Teksten en 2 opdrachten (worden uitgedeeld) 1. Intuïtieve benadering 2. Formele benadering

34 Voorbeeldprobleem Gegeven: In Orange County VS is 51 % man, 9.5% van de mannen rookt sigaren, tegenover 1.7% van de vrouwen Gevraagd: Hoe groot is de kans dat een willekeurige sigarenroker een man is?

35 1. Intuïtieve benadering

36 2. Formele benadering

37 Einde vooruitblik Volgende week (week 6): –Vragenuur –Geen nieuwe stof –Voorbereiding proeftentamen


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