Chapter 2: ANOVA and regression. Caroline Verhoeven
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1 Chapter 2: ANOVA and regression Caroline Verhoeven
2 Table of contents 1 ANOVA One-way ANOVA Repeated measures ANOVA Two-way ANOVA 2 Regression Simple linear regression Multiple regression 3 Conclusion Caroline Verhoeven BMOL-G / 29
3 1. ANOVA 1. One-way ANOVA One-way ANOVA We want to compare more than two groups Example : We want to compare the effect of treatment 1, treatment 2 and a placebo H 0 : µ 1 = µ 2 = = µ k H a : at least one of the means is different Caroline Verhoeven BMOL-G / 29
4 1. ANOVA 1. One-way ANOVA One-way ANOVA We want to compare more than two groups Example : We want to compare the effect of treatment 1, treatment 2 and a placebo H 0 : µ 1 = µ 2 = = µ k H a : at least one of the means is different Principle : Compare the variation of the means between the groups to the variation inside the groups. In SPSS : Analyse Comparer les moyennes ANOVA à 1 facteur (Analyze Compare Means One-Way ANOVA) Caroline Verhoeven BMOL-G / 29
5 Conditions 1. ANOVA 1. One-way ANOVA We must have simple random samples The samples must be independent σ1 2 = σ2 2 = = σ2 k We can use the Levene test (in SPSS) The variables must be normally distributed Caroline Verhoeven BMOL-G / 29
6 Conditions 1. ANOVA 1. One-way ANOVA We must have simple random samples The samples must be independent σ1 2 = σ2 2 = = σ2 k We can use the Levene test (in SPSS) The variables must be normally distributed If not : Kruskal-Wallis test Done in the same way as the Mann-Whitney test Caroline Verhoeven BMOL-G / 29
7 Preliminary test : Levene test 1. ANOVA 1. One-way ANOVA Are the variances the same for all the groups? Caroline Verhoeven BMOL-G / 29
8 Preliminary test : Levene test 1. ANOVA 1. One-way ANOVA Are the variances the same for all the groups? Do the Levene test : H 0 : σ 2 1 = σ2 2 = = σ2 k H a : At least 1 of the variances is different Caroline Verhoeven BMOL-G / 29
9 Preliminary test : Levene test 1. ANOVA 1. One-way ANOVA Are the variances the same for all the groups? Do the Levene test : H 0 : σ 2 1 = σ2 2 = = σ2 k H a : At least 1 of the variances is different p > 0, 05 : NRH 0 p < 0, 05 : RH 0 Caroline Verhoeven BMOL-G / 29
10 Preliminary test : Levene test 1. ANOVA 1. One-way ANOVA Are the variances the same for all the groups? Do the Levene test : H 0 : σ 2 1 = σ2 2 = = σ2 k H a : At least 1 of the variances is different p > 0, 05 : NRH 0 p < 0, 05 : RH 0 SPSS : ANOVA Options test d homogénéité des variances (ANOVA Options Homogeneity of variance test) Caroline Verhoeven BMOL-G / 29
11 1. ANOVA 1. One-way ANOVA what with a RH 0 for the Levene test? What do we do if we have a RH 0 for the Levene test? Caroline Verhoeven BMOL-G / 29
12 1. ANOVA 1. One-way ANOVA what with a RH 0 for the Levene test? What do we do if we have a RH 0 for the Levene test? A solution : the Welch test Caroline Verhoeven BMOL-G / 29
13 1. ANOVA 1. One-way ANOVA what with a RH 0 for the Levene test? What do we do if we have a RH 0 for the Levene test? A solution : the Welch test SPSS : ANOVA Options Welch Caroline Verhoeven BMOL-G / 29
14 After ANOVA 1. ANOVA 1. One-way ANOVA What to do when we have a RH 0 for the ANOVA? How can we know which means are different? Caroline Verhoeven BMOL-G / 29
15 After ANOVA 1. ANOVA 1. One-way ANOVA What to do when we have a RH 0 for the ANOVA? How can we know which means are different? If we have a NRH 0 : stop If we have a RH 0 : there are different multiple comparison tests Caroline Verhoeven BMOL-G / 29
16 After ANOVA 1. ANOVA 1. One-way ANOVA What to do when we have a RH 0 for the ANOVA? How can we know which means are different? If we have a NRH 0 : stop If we have a RH 0 : there are different multiple comparison tests Bonferroni Sidak Tukey Dunnett... Caroline Verhoeven BMOL-G / 29
17 1. ANOVA 1. One-way ANOVA After ANOVA What to do when we have a RH 0 for the ANOVA? How can we know which means are different? If we have a NRH 0 : stop If we have a RH 0 : there are different multiple comparison tests Bonferroni Sidak Tukey Dunnett... SPSS : ANOVA Post Hoc Caroline Verhoeven BMOL-G / 29
18 Principle 1. ANOVA 2. Repeated measures ANOVA A measurement is taken in k different situations, on the same subjects Caroline Verhoeven BMOL-G / 29
19 1. ANOVA 2. Repeated measures ANOVA Principle A measurement is taken in k different situations, on the same subjects Generalization of the t-test for 2 paired samples Caroline Verhoeven BMOL-G / 29
20 1. ANOVA 2. Repeated measures ANOVA Principle A measurement is taken in k different situations, on the same subjects Generalization of the t-test for 2 paired samples SPSS : Analyse Modèle Linéaire Général Mesures Répétées (Analyze General Linear Model Repeated Measures) Caroline Verhoeven BMOL-G / 29
21 1. ANOVA 2. Repeated measures ANOVA Conditions We must have a simple random sample the variances of the differences between two groups must be equal we can use the Mauchly test The variables must be normally distributed Caroline Verhoeven BMOL-G / 29
22 1. ANOVA 2. Repeated measures ANOVA Conditions We must have a simple random sample the variances of the differences between two groups must be equal we can use the Mauchly test The variables must be normally distributed If not : Friedman test Done in the same way as the signed rank Wilcoxon test Caroline Verhoeven BMOL-G / 29
23 Preliminary test : Mauchly test 1. ANOVA 2. Repeated measures ANOVA Are the variances of the differences between two groups equal? Caroline Verhoeven BMOL-G / 29
24 Preliminary test : Mauchly test 1. ANOVA 2. Repeated measures ANOVA Are the variances of the differences between two groups equal? Perform the Mauchly test H 0 : The variances of the differences are equal H a : At least 1 variance is different Caroline Verhoeven BMOL-G / 29
25 Preliminary test : Mauchly test 1. ANOVA 2. Repeated measures ANOVA Are the variances of the differences between two groups equal? Perform the Mauchly test H 0 : The variances of the differences are equal H a : At least 1 variance is different p > 0, 05 : NRH 0 p < 0, 05 : RH 0 Caroline Verhoeven BMOL-G / 29
26 1. ANOVA 2. Repeated measures ANOVA Preliminary test : Mauchly test Are the variances of the differences between two groups equal? Perform the Mauchly test H 0 : The variances of the differences are equal H a : At least 1 variance is different p > 0, 05 : NRH 0 p < 0, 05 : RH 0 If RH 0 : Greenhouse-Geisser correction (stronger correction) Huynh-Feldt correction Caroline Verhoeven BMOL-G / 29
27 1. ANOVA 2. Repeated measures ANOVA Preliminary test : Mauchly test Are the variances of the differences between two groups equal? Perform the Mauchly test H 0 : The variances of the differences are equal H a : At least 1 variance is different p > 0, 05 : NRH 0 p < 0, 05 : RH 0 If RH 0 : Greenhouse-Geisser correction (stronger correction) Huynh-Feldt correction To choose : look at the Greenhouse-Geisser epsilon Caroline Verhoeven BMOL-G / 29
28 Two-Way ANOVA 1. ANOVA 3. Two-way ANOVA We want to evaluate the effects of 2 factors Example : We want to study the white blood cells level for persons with and without leukemia and for children and adults Caroline Verhoeven BMOL-G / 29
29 Two-Way ANOVA 1. ANOVA 3. Two-way ANOVA We want to evaluate the effects of 2 factors Example : We want to study the white blood cells level for persons with and without leukemia and for children and adults 2 factors possibility of interaction Caroline Verhoeven BMOL-G / 29
30 1. ANOVA 3. Two-way ANOVA Two-Way ANOVA We want to evaluate the effects of 2 factors Example : We want to study the white blood cells level for persons with and without leukemia and for children and adults Formulation of the null hypotheses : 2 factors possibility of interaction H 0 : having leukemia or not makes no difference wrt the white blood cells levels H 0 : being a child or an adult makes no difference wrt the white blood cells levels H 0 : The effect of the illness on the white blood cells levels does not depend on being a child or an adult Caroline Verhoeven BMOL-G / 29
31 1. ANOVA 3. Two-way ANOVA Two-Way ANOVA We want to evaluate the effects of 2 factors Example : We want to study the white blood cells level for persons with and without leukemia and for children and adults Formulation of the null hypotheses : 2 factors possibility of interaction H 0 : having leukemia or not makes no difference wrt the white blood cells levels H 0 : being a child or an adult makes no difference wrt the white blood cells levels H 0 : The effect of the illness on the white blood cells levels does not depend on being a child or an adult SPSS :Analyse Modèle Linéaire Général Univarié (Analyze General Linear Model Univariate) Caroline Verhoeven BMOL-G / 29
32 2. Regression 1. Simple linear regression Linear regression x i : quantitative data, predictor y i : quantitative data, response Caroline Verhoeven BMOL-G / 29
33 2. Regression 1. Simple linear regression Linear regression x i : quantitative data, predictor y i : quantitative data, response Linear relationship between X and Y Caroline Verhoeven BMOL-G / 29
34 2. Regression 1. Simple linear regression Linear regression x i : quantitative data, predictor y i : quantitative data, response Linear relationship between X and Y Question : How do we determine the regression line y = b 0 + b 1 x, b 0? b 1? Caroline Verhoeven BMOL-G / 29
35 Linear regression : conditions 2. Regression 1. Simple linear regression Due to the measurement error and biological variability, we have : y i = b 0 + b 1 x i + ε i ε i : residue, condition : ε i N (0, σ 2 ) Caroline Verhoeven BMOL-G / 29
36 Linear regression : conditions 2. Regression 1. Simple linear regression Due to the measurement error and biological variability, we have : y i = b 0 + b 1 x i + ε i ε i : residue, condition : ε i N (0, σ 2 ) σ : independent of x Homocedasticity Age FCM Heterocedasticity Age Caroline Verhoeven BMOL-G / 29
37 Multiple regression : utility 2. Regression 2. Multiple regression A variable can depend simultaneously of several factors. Caroline Verhoeven BMOL-G / 29
38 2. Regression 2. Multiple regression Multiple regression : utility A variable can depend simultaneously of several factors. Exemple 1 Prediction of the height of a person from the height of his mother, his father and his gender Caroline Verhoeven BMOL-G / 29
39 2. Regression 2. Multiple regression Multiple regression : utility A variable can depend simultaneously of several factors. Exemple 1 Prediction of the height of a person from the height of his mother, his father and his gender Goal : Predict the values of Y from several variables X 1, X 2,..., X k Caroline Verhoeven BMOL-G / 29
40 2. Regression 2. Multiple regression Multiple regression : utility A variable can depend simultaneously of several factors. Exemple 1 Prediction of the height of a person from the height of his mother, his father and his gender Goal : Predict the values of Y from several variables X 1, X 2,..., X k X 1, X 2,...X k are generally quantitative or ordinal, a few can be nominal Caroline Verhoeven BMOL-G / 29
41 2. Regression 2. Multiple regression Multiple regression : utility A variable can depend simultaneously of several factors. Exemple 1 Prediction of the height of a person from the height of his mother, his father and his gender Goal : Predict the values of Y from several variables X 1, X 2,..., X k X 1, X 2,...X k are generally quantitative or ordinal, a few can be nominal Exemple 1 Gender is a nominal variable : 0=femme 1=homme Caroline Verhoeven BMOL-G / 29
42 2. Regression 2. Multiple regression Multiple regression : principle Study of the linear relationship between Y and variables X 1, X 2,..., X k : y = b 0 + b 1 x 1 + b 2 x b k x k. Caroline Verhoeven BMOL-G / 29
43 2. Regression 2. Multiple regression Multiple regression : principle Study of the linear relationship between Y and variables X 1, X 2,..., X k : y = b 0 + b 1 x 1 + b 2 x b k x k. looking for : b 0, b 1, b 2,..., b k SPSS : Analyse Régression Linéaire (Analyze Regression Linear) Caroline Verhoeven BMOL-G / 29
44 2. Regression 2. Multiple regression Methods for the regression I Enter Method : Method who introduces all the independent variables simultaneously Use it if you want to keep all the independent variabls int he regression equation. Caroline Verhoeven BMOL-G / 29
45 2. Regression 2. Multiple regression Methods for the regression I Enter Method : Method who introduces all the independent variables simultaneously Use it if you want to keep all the independent variabls int he regression equation. The other methods are hierarchal methods. Only use it if you think that some variables are more important than others Caroline Verhoeven BMOL-G / 29
46 Methods for the regression II 2. Regression 2. Multiple regression Forward method : Introduction of an independent variable at the time Order : decreasing order of correlation coefficient between dependent and independent variable If it doesn t enhance the model significantly, it is eliminated We stop at the first eliminated variable Caroline Verhoeven BMOL-G / 29
47 Methods for the regression II 2. Regression 2. Multiple regression Forward method : Introduction of an independent variable at the time Order : decreasing order of correlation coefficient between dependent and independent variable If it doesn t enhance the model significantly, it is eliminated We stop at the first eliminated variable Backward method All the independent variables are introduced The weakest variable is removed If the model is significantly worse, it is reintroduced We stop at the first reintroduced variable Caroline Verhoeven BMOL-G / 29
48 Methods for the regression II 2. Regression 2. Multiple regression Forward method : Introduction of an independent variable at the time Order : decreasing order of correlation coefficient between dependent and independent variable If it doesn t enhance the model significantly, it is eliminated We stop at the first eliminated variable Backward method All the independent variables are introduced The weakest variable is removed If the model is significantly worse, it is reintroduced We stop at the first reintroduced variable Stepwize method Introduction of an independent variable at the time Testing if the new variables, and the previous, are significant Non significant variables are eliminated Caroline Verhoeven BMOL-G / 29
49 2. Regression 2. Multiple regression Multiple regression : conditions 1 The same conditions as for a simple regression 2 There exists a linear relationship between Y and the X i 3 No multi-collinearity : X i variables must not be highly correlated 4 You need a lot of subjects. Minimum : 5k, k : number of factors Caroline Verhoeven BMOL-G / 29
50 3. Conclusion Use of ANOVA and regression Model Response Predictor One-way ANOVA 1 quantitative 1 qualitative Two-way ANOVA 1 quantitative 2 qualitative Simple regression 1 quantitative 1 quantitative Multiple regression 1 quantitative 2 or more quantitative Caroline Verhoeven BMOL-G / 29
51 4. Exercises Exercise 1 Open the file film.xls. Determine the regression equation of the income of a movie based on a book : Production costs Publicity costs Income of the book 10 movies are considered. Caroline Verhoeven BMOL-G / 29
52 Exercise 2 4. Exercises Open the file melatonin.sav in SPSS. Traveling to another timezone induces a jet lag. We re adapting gradually because our circadian rhythm is synchronized with the day-night cycle. This rhythm change is called a shift En 2002, Wright et Czeisler studied this phenomenon. The time of melatonin production for 22 subjects. Those subjects were randomly divided in 3 groups. Alle the groups were waken 3 hours during their sleep. The first group with light in the eyes, the second with lights behind the knee, while there was no light at all for the third group. Their meltatonin cycle was measure before the treatment and after 2 days. Caroline Verhoeven BMOL-G / 29
53 4. Exercises Exercise 2 Open the file melatonin.sav in SPSS. Traveling to another timezone induces a jet lag. We re adapting gradually because our circadian rhythm is synchronized with the day-night cycle. This rhythm change is called a shift En 2002, Wright et Czeisler studied this phenomenon. The time of melatonin production for 22 subjects. Those subjects were randomly divided in 3 groups. Alle the groups were waken 3 hours during their sleep. The first group with light in the eyes, the second with lights behind the knee, while there was no light at all for the third group. Their meltatonin cycle was measure before the treatment and after 2 days. The shift (in hours) is given in the file. A negative shifts corresponds to a delay. Determine if the type of lightning has an effect on the circadian cycle. Caroline Verhoeven BMOL-G / 29
54 Exercise 3 4. Exercises In 2005, Walker et al. studied stress among penguins. Some of those penguins reproduce in quite regions, while others reproduce in touristic regions. Caroline Verhoeven BMOL-G / 29
55 Exercise 3 4. Exercises In 2005, Walker et al. studied stress among penguins. Some of those penguins reproduce in quite regions, while others reproduce in touristic regions. We want to know how stress of young penguins is impacted by their age and by the type of region in which they re growing. Caroline Verhoeven BMOL-G / 29
56 4. Exercises Exercise 3 In 2005, Walker et al. studied stress among penguins. Some of those penguins reproduce in quite regions, while others reproduce in touristic regions. We want to know how stress of young penguins is impacted by their age and by the type of region in which they re growing. To answer this question, young penguins were captured and their corticosterone level was measured. This was done in both regions, with penguins which just hatched, which were 40 to 50 days old and with young adults. Caroline Verhoeven BMOL-G / 29
57 4. Exercises Exercise 3 In 2005, Walker et al. studied stress among penguins. Some of those penguins reproduce in quite regions, while others reproduce in touristic regions. We want to know how stress of young penguins is impacted by their age and by the type of region in which they re growing. To answer this question, young penguins were captured and their corticosterone level was measured. This was done in both regions, with penguins which just hatched, which were 40 to 50 days old and with young adults. Open the file penguin.xls. Determine if the age and the region have an impact on the stress level of the penguins. Caroline Verhoeven BMOL-G / 29
58 Exercise 4 4. Exercises Open the file intima media.xls in SPSS. 1 Do age and BMI influence significantly the thickness of the intima-media. 2 Is there a significant difference for the thickness of the intima-media depending on the consumption of tobacco and alcohol? Caroline Verhoeven BMOL-G / 29
59 Exercise 5 4. Exercises In the reality-show I m a celebrity, get me out of here, celebrities have to survive in the jungle and have to undergo humiliating tests. One of those tests is to swallow unappetizing food. 8 of those celebrities have to swallow 4 items each. The time between the moment they put the food item in the mouth, and which they feel sick is measured in seconds. The results can be found in the file celeb.xls. 1 Do we have different times for the different food items? Caroline Verhoeven BMOL-G / 29
60 Exercise 6 4. Exercises Mammals have mechanisms to reduce the brain temperature with respect to the body temperature when exposed to a long period of extreme heat. In 2003 Fuller et al did an experience on ostriches to see if they can do the same things. The results can be found in the file ostrich.xls. Do the brain and body temperature differ for ostriches? Caroline Verhoeven BMOL-G / 29
61 Exercise 7 4. Exercises We are testing the influence of various diets on lab rats. The weight gain of the rats can be found in the variable Weight expressed in grams. The value for the variable Calorie is 1 if the rats had a normal diet. The value is 2 if they had an hypercaloric diet. The value of the variable Vitamin is 1 if the rats didn t receive a vitamin supplement. The value is 2 if they received such a supplement. Is the wight gain of the rats influenced by the diet and the vitamin supplement? The data are in the file rats.xls. Caroline Verhoeven BMOL-G / 29
62 Exercise 8 4. Exercises The alarm call of a chickadee for a not flying predator sounds like dee-dee-dee. In 2005, Templeton et al. wanted to know if the number of dee is correlated with the weight of te predator. They perched 13 predators of different sizes before a flock of chicakdees. The average number of dee was measured for each predator. The data can be found in the file chickadee.xls. Can the weight of the predator be predicted from the average number of dee? Caroline Verhoeven BMOL-G / 29
63 4. Exercises Exercises on children with the down syndrome I In 2013, de Graaf et al. studied the importance of the choice of education system (specialized education or traditional education) for children with down syndrome. They asked parents with children with the down syndrome to fill a survey. The questions were about the performances of the children in reading, writing and maths. They also for the age and the IQ of the cild and the study level of the parents. The data can be found in the file down.xls Caroline Verhoeven BMOL-G / 29
64 4. Exercises Exercises on children with the down syndrome II Exercise 9 Is there a difference in performances in reading, writing and maths between children in the traditional and the specialized education system. Exercise 10 Is the IQ different for children in the traditional and specialized education system? Exercise 11 What is the effect of the study level of the parents on the performances of the child in writing. Exercise 12 What is the effect of the IQ and of the age of the child and the study level of the mother with respect to the performance of the child in maths. Exercice 13 What is the impact of the study level of the parents on the choice of the education system. Caroline Verhoeven BMOL-G / 29
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