Analysis of Variance

Size: px
Start display at page:

Download "Analysis of Variance"

Transcription

1 Analysis of Variance Blood coagulation time T avg A B C D

2 Blood coagulation time A B C D Combined Coagulation Time Notation Assume we have k treatment groups. n t N Y ti Ȳ t Ȳ number of subjects in treatment group t number of subjects (overall) response i in treatment group t average response in treatment group t average response (overall)

3 Variance contributions (Y ti Ȳ)2 = t t i n t (Ȳt Ȳ)2 + t (Y ti Ȳt ) 2 i S T = S B + S W N 1 = k 1 + N k Estimating the variability We assume that the data are random samples from four normal distributions having the same variance σ 2, differing only (if at all) in their means. We can estimate the variance σ 2 for each treatment t, using the sum of squared differences from the averages within each group. Define, for treatment group t, Then n t S t = (Y ti Ȳt ) 2. i = 1 E(S t )=(n t 1) σ 2.

4 Within group variability The within-group sum of squares is the sum of all treatment sum of squares: S W =S S k = t (Y ti Ȳt ) 2 i The within-group mean square is defined as S S k M W = (n 1 1)+ +(n k 1) = S W N k = t i (Y ti Ȳt ) 2 N k It is our first estimate of σ 2. Between group variability The between-group sum of squares is k S B = n t (Ȳt Ȳ)2 t = 1 The between-group mean square is defined as It is our second estimate of σ 2. That is, if there is no treatment effect! M B = S B t k 1 = n t(ȳt Ȳ)2 k 1

5 Important facts The following are facts that we will exploit later for some formal hypothesis testing: The distribution of S W /σ 2 is χ 2 (df=n-k) The distribution of S B /σ 2 is χ 2 (df=k-1) if there is no treatment effect! S W and S B are independent The F distribution Let Z 1 χ 2 m, and Z 2 χ 2 n. Assume Z 1 and Z 2 are independent. Then Z 1 /m Z 2 /n F m,n F distributions df=20,10 df=20,20 df=20,

6 ANOVA table source sum of squares df mean square between treatments within treatments total S B = t S W = t S T = t n t (Ȳt Ȳ)2 k 1 M B =S B /(k 1) (Y ti Ȳt ) 2 N k M W =S W /(N k) i (Y ti Ȳ)2 N 1 i Example source sum of squares df mean square between treatments within treatments total

7 The ANOVA model We write Y ti = µ t + ɛ ti with ɛ ti iid N(0,σ 2 ). Using τ t = µ t µ we can also write Y ti = µ + τ t + ɛ ti. The corresponding analysis of the data is y ti = ȳ + (ȳ t ȳ ) + (y ti ȳ t ) The ANOVA model Three different ways to describe the model: A. Y ti independent with Y ti N(µ t, σ 2 ) B. Y ti = µ t + ɛ ti where ɛ ti iid N(0, σ 2 ) C. Y ti = µ + τ t + ɛ ti where ɛ ti iid N(0, σ 2 ) and t τ t = 0

8 Now what did we do...? = observations grand average treatment deviations residuals y ti = ȳ + ȳ t ȳ + y ti ȳ t Vector Y = A + T + R Sum of Squares 98,644 = 98, D s of Freedom 24 = Hypothesis testing We assume Y ti = µ + τ t + ɛ ti with ɛ ti iid N(0,σ 2 ). Equivalently, Y ti independent N(µ t, σ 2 ) We want to test H 0 : τ 1 = =τ k =0 versus H a : H 0 is false. Equivalently, H 0 : µ 1 =... =µ k For this, we use a one-sided F test.

9 Another fact It can be shown that E(M B )=σ 2 + t n tτ 2 t k 1 Therefore E(M B )=σ 2 E(M B ) > σ 2 if H 0 is true if H 0 is false Recipe for the hypothesis test Under H 0 we have Therefore Calculate M B and M W. Calculate M B /M W. M B M W F k 1, N k. Calculate a p-value using M B /M W as test statistic, using the right tail of an F distribution with k 1 and N k degrees of freedom.

10 Example (cont) H 0 : τ 1 =τ 2 =τ 3 =τ 4 =0 versus H a : H 0 is false. M B = 76, M W =5.6, therefore M B /M W = Using an F distribution with 3 and 20 degrees of freedom, we get a pretty darn low p-value. Therefore, we reject the null hypothesis. F(3,20) M B M W Another example A B treatment response Are the population means the same? By now, we know two ways of testing that: Two-sample t-test, and ANOVA with two treatments. But do they give similar results?

11 ANOVA table source sum of squares df mean square between treatments within treatments total S B = t S W = t S T = t n t (Ȳt Ȳ)2 k 1 M B =S B /(k 1) (Y ti Ȳt ) 2 N k M W =S W /(N k) i (Y ti Ȳ)2 (N 1) i ANOVA for two groups The ANOVA test statistic is M B /M W, with M B =n 1 (Ȳ1 Ȳ)2 + n 2 (Ȳ2 Ȳ)2 and M W = n1 i = 1 (Y 1i Ȳ1) 2 + n 2 i = 1 (Y 2i Ȳ2) 2 n 1 + n 2 2

12 Two-sample t-test The test statistic for the two sample t-test is Ȳ 1 t= Ȳ2 s 1/n 1 + 1/n 2 with n1 s 2 i = 1 = (Y 1i Ȳ1) 2 + n 2 i = 1 (Y 2i Ȳ2) 2 n 1 + n 2 2 This also assumes equal variance within the groups! Reference distributions Result: M B M W =t 2 If there was no difference in means, then M B M W F 1,n1 +n 2 2 t t n1 +n 2 2 Now does this mean F 1,n1 +n 2 2=(t n1 +n 2 2) 2?

13 A few facts F 1,k =t 2 k F k, = χ2 k k N(0,1) 2 = χ 2 1 =F 1, =t 2 Fixed effects Underlying group dist ns µ 8 Standard ANOVA model µ 7 Data µ 6 µ 5 µ 4 µ 3 µ 2 µ 1

14 Random effects Dist n of group means Underlying group dist ns µ 8 Random effects model µ µ 7 Data Observed underlying group means µ 6 µ 5 µ 4 µ 3 µ 2 µ 1 The random effects model Two different ways to describe the model: A. µ t iid N(µ, σ 2 A ) Y ti = µ t + ɛ ti where ɛ ti iid N(0, σ 2 ) B. τ t iid N(0, σ 2 A ) Y ti = µ + τ t + ɛ ti where ɛ ti iid N(0, σ 2 ) We add another layer of sampling.

15 Hypothesis testing In the standard ANOVA model, we considered the µ t as fixed but unknown quantities. We test the hypothesis H 0 : µ 1 = = µ k (versus H 0 is false) using the statistic M B /M W from the ANOVA table and the comparing this to an F(k 1, N k) distribution. In the random effects model, we consider the µ t as random draws from a normal distribution with mean µ and variance σ 2 A. We seek to test the hypothesis H 0 : σ 2 A = 0 versus H a : σ 2 A > 0. As it turns out, we end up with the same test statistic and same null distribution. For one-way ANOVA, that is! Estimation For the random effects model it can be shown that E(M B )=σ 2 +n 0 σ 2 A where n 0 = 1 ( ) N t n2 t k 1 t n t Recall also that E(M W )=σ 2. Thus, we may estimate σ 2 by ˆσ 2 = M W. And we may estimate σ 2 A by ˆσ2 A =(M B M W )/n 0 (provided that this is 0).

16 Random effects example Subject ID response Random effects example The samples sizes for the 8 subjects were (14, 12, 11, 10, 10, 11, 15, 9), for a total sample size of 92. Thus, n source SS df MS F P-value between subjects within subjects total We have M B = 212 and M W = 46. Thus ˆσ = 46 = 6.8 overall sample mean = 40.3 ˆσ A = (212 46)/11.45 = 3.81.

17 ANOVA assumptions Data in each group are a random sample from some population. Observations within groups are independent. Samples are independent. Underlying populations normally distributed. Underlying populations have the same variance. The Kruskal-Wallis test is a non-parametric rank-based approach to assess differences in means. In the case of two groups, the Kruskal-Wallis test reduces exactly to the Wilcoxon rank-sum test. This is just like how ANOVA with two groups is equivalent to the two-sample t test. Multiple comparisons When we carry out an ANOVA on k treatments, we test H 0 : µ 1 = =µ k versus H a : H 0 is false Assume we reject the null hypothesis, i.e. we have some evidence that not all treatment means are equal. Then we could for example be interested in which ones are the same, and which ones differ. For this, we might have to carry out some more hypothesis tests. This procedure is referred to as multiple comparisons.

18 Key issue We will be conducting, say, T different tests, and we become concerned about the overall error rate (sometimes called the familywise error rate). Overall error rate=pr( reject at least one H 0 all H 0 are true ) =1 {1 Pr( reject first first H 0 is true )} T if independent T Pr( reject first first H 0 is true ) generally Types of multiple comparisons There are two different types of multiple comparisons procedures: Sometimes we already know in advance what questions we want to answer. Those comparisons are called planned (or a priori) comparisons. Sometimes we do not know in advance what questions we want to answer, and the judgement about which group means will be studied the same depends on the ANOVA outcome. Those comparisons are called unplanned (or a posteriori) comparisons.

19 Former example We previously investigated whether the mean blood coagulation times for subjects receiving different treatments (A, B, C or D) were the same. Imagine A is the standard treatment, and we wish to compare each of treatments B, C, D to treatment A. planned comparisons! After inspecting the treatment means, we find that A and D look similar, and B and C look similar, but A and D are quite different from B and C. We might want to formally test the hypothesis µ A =µ D µ B =µ C. unplanned comparisons! Adjusting the significance level Assume the investigator plans to make T independent significance tests, all at the significance level α. If all the null hypothesis are true, the probability of making no Type I error is (1 α ) T. Hence the overall significance level is α=1 (1 α ) T Solving the above equation for α yields α =1 (1 α) 1 T The above adjustment is called the Dunn Sidak method.

20 An alternative method In the literature, investigators often use α = α T where T is the number of planned comparisons. This adjustment is called the Bonferroni method. Unplanned comparisons Suppose we are comparing k treatment groups. Suppose ANOVA indicates that you reject H 0 : µ 1 = = µ k What next? Which of the µ s are different from which others? Consider testing H 0 : µ i = µ j for all pairs i,j. There are ( ) k 2 = k (k 1) 2 such pairs. k = 5 ( k 2) = 10. k = 10 ( k 2) = 45.

21 Bonferroni correction Suppose we have 10 treatment groups, and so 45 pairs. If we perform 45 t-tests at the significance level α = 0.05, we would expect to reject 5% 45 2 of them, even if all of the means were the same. Let α = Pr(reject at least one pairwise test all µ s the same) (no. tests) Pr(reject test #1 µ s the same) The Bonferroni correction: Use α = α/(no. tests) as the significance level for each test. For example, with 10 groups and so 45 pairwise tests, we would use α = 0.05 / for each test. Blood coagulation time A B C D Combined Coagulation Time

22 Pairwise comparisons Comparison p-value α = α k = = A vs B A vs C < A vs D B vs C B vs D < C vs D < Another example S A treatment F G C response

23 ANOVA table Source SS Df MS F-value p-value Between treatment < Within treatment ( 5 ) 2 = 10 pairwise comparisons α = 0.05/10 = For each pair, consider T i,j = ( ) ( ) Ȳ i Ȳj / ˆσ ni nj Use ˆσ = M W (M W = within-group mean square) and refer to a t distribution with df = 45. A comparison Uncorrected: Each interval, individually, had (in advance) a 95% chance of covering the true mean difference. Corrected: A : S G : S G : A F : S F : A F : G C : S C : A Uncorrected Bonferroni Tukey C : G C : F (In advance) there was a greater than 95% chance that all of the intervals would cover their respective parameters.!10! Difference in response

24 Newman-Keuls procedure Goal: Identify sets of treatments whose mean responses are not significantly different. (Assuming equal sample sizes for the treatment groups.) Procedure: 1. Calculate the group sample means. 2. Order the sample means from smallest to largest. 3. Calculate a triangular table of all pairwise sample means. 4. Calculate q i = Q α (i, df) for i = 2, 3,..., k. The Q is called the studentized range distribution! 5. Calculate R i = q i M W /n. Newman-Keuls procedure (continued) Procedure: 6. If the difference between the biggest and the smallest means is less than R k, draw a line under all of the means and stop. 7. Compare the second biggest and the smallest (and the second-smallest and the biggest) to R k 1. If observed difference is smaller than the critical value, draw a line between these means. 8. Continue to look at means for which a line connecting them has not yet been drawn, comparing the difference to R i with progressively smaller i s.

25 Example Sorted sample means: A F G S C Table of differences: F G S C A F G S 6.0 Example (continued) From the ANOVA table: M W = 5.46 n = 10 for each group MW /10 = df = 45 The q i (using df=45 and α = 0.05): q 2 q 3 q 4 q R i =q i M W /10: R 2 R 3 R 4 R

26 Example (continued) Table of differences: F G S C A F G S 6.0 R i =q i M W /10: R 2 R 3 R 4 R Results Sorted sample means: A F G S C Interpretation: A F G < S < C

27 Another example Sorted sample means: D C A B E Interpretation: {D, C, A, B} < E and D < {A, B} Nested ANOVA: Example We have: 3 hospitals 4 subjects within each hospital 2 independent measurements per subject Hospital I Hospital II Hospital III

28 The model Hospitals Hospitals!30!20! Individuals!30!20! Individuals!30!20! Residuals!30!20! Residuals Nested ANOVA: models Y ijk = µ + α i + β ij + ɛ ijk µ = overall mean α i = effect for ith hospital β ij = effect for jth subject within ith hospital ɛ ijk = random error Random effects model α i Normal(0, σa 2 ) β ij Normal(0, σb A 2 ) ɛ ijk Normal(0, σ 2 ) Mixed effects model α i fixed; α i =0 β ij Normal(0, σb A 2 ) ɛ ijk Normal(0, σ 2 )

29 Example: sample means Hospital I Hospital II Hospital III Ȳ ij Ȳ i Ȳ Calculations (equal sample sizes) Source Sum of squares df among groups SS among =bn i (Ȳi Ȳ ) 2 a 1 subgroups within groups within subgroups TOTAL SS subgr =n i j (Ȳij Ȳi ) 2 a (b 1) SS within = i j k (Y ijk Ȳij ) 2 a b (n 1) i j k (Y ijk Ȳ ) 2 a b n 1

30 ANOVA table SS df MS F expected MS SS among a 1 SS among a 1 MS among MS subgr σ 2 + n σ 2 B A + nbσ2 A SS subgr a (b 1) SS subgr a(b 1) MS subgr MS within σ 2 + n σ 2 B A SS within a b (n 1) SS within ab(n 1) σ 2 SS total a b n 1 Example source df SS MS F P-value among groups among subgroups within groups < within subgroups TOTAL

31 Variance components Within subgroups (error; between measurements on each subject) s 2 =MS within =1.30 s = 1.30 = 1.14 Among subgroups within groups (among subjects within hospitals) s 2 B A =MS subgr MS within = =94.94 s B A = = 9.74 n 2 Among groups (among hospitals) s 2 A =MS among MS subgr = =17.71 s A = = 4.21 nb 8 Variance components (2) s 2 + s 2 B A + s2 A = = s 2 s 2 B A represents = 1.1% represents = 83.3% s 2 A represents = 15.6% Note: var(y) = σ 2 + σ 2 B A + σ2 A var(y A) = σ 2 + σ 2 B A var(y A, B) = σ 2

32 Subject averages I-1 I-2 I-3 I-4 II-1 II-2 II-3 II-4 III-1 III-2 III-3 III ave ANOVA table source df SS MS F P-value between within Higher-level nested ANOVA models You can have as many levels as you like. For example, here is a three-level nested mixed ANOVA model: Y ijkl =µ + α i + B ij + C ijk + ɛ ijkl Assumptions: B ij N(0,σ 2 B A ), C ijk N(0,σ 2 C B ), ɛ ijkl N(0,σ 2 ).

33 Calculations Source Sum of squares df among groups SS among =b c n i (Ȳi Ȳ ) 2 a 1 among subgroups among subsubgroups within subsubgroups SS subgr =c n i j (Ȳij Ȳi ) 2 a (b 1) j SS subsubgr =n i SS subsubgr = i j k (Ȳijk Ȳij ) 2 a b (c 1) l (Y ijkl Ȳijk ) 2 a b c (n 1) k ANOVA table SS MS F expected MS SS among bcn a (ȲA Ȳ)2 a 1 MS among MS subgr α σ 2 + nσc B ncσ2 B A + ncb a 1 SS subgr cn a b (ȲB ȲA) 2 a(b 1) MS subgr MS subsubgr σ 2 + nσ 2 C B + ncσ2 B A SS subsubgr n a b c (ȲC ȲB) 2 ab(c 1) MS subsubgr MS within σ 2 + nσ 2 C B SS within 2 a b c n (Y ȲC) abc(n 1) σ 2

34 Unequal sample size It is best to design your studies such that you have equal sample sizes in each cell. However, once in a while this is not possible. In the case of unequal sample sizes, the calculations become really painful (though a computer can do all of the calculations for you). Even worse, the F tests for the upper levels in the ANOVA table no longer have a clear null distribution. Maximum likelihood methods are more complicated, but can solve this problem. Two-way ANOVA Treatment Gender Male Female Let r c n be the number of rows in the two-way ANOVA, be the number of columns in the two-way ANOVA, be the number of observations within each of those r c groups.

35 A picture Response Female Male Female Male All sorts of means Treatment Gender 1 2 Male Female This table shows the cell, row, and column means, plus the overall mean.

36 Two-way ANOVA table source sum of squares df between rows SS rows =c n i (Ȳi Ȳ ) 2 r 1 between columns SS columns =r n j (Ȳ j Ȳ ) 2 c 1 interaction SS interaction (r 1)(c 1) error SS within = i j k (Y ijk Ȳij ) 2 rc(n 1) total SS total = i j k (Y ijk Ȳ ) 2 rcn 1 Example source sum of squares df mean squares sex treatment interaction error

37 The ANOVA model Let Y ijk be the k th item in the subgroup representing the i th group of factor A (r levels) and the j th group of factor B (c levels). We write Y ijk =µ + α i + β j + γ ij + ɛ ijk The corresponding analysis of the data is y ijk = ȳ + (ȳ i ȳ ) + (ȳ j ȳ ) + (ȳ ij ȳ i ȳ j + ȳ ) + (y ijk ȳ ij ) Towards hypothesis testing source mean squares expected mean squares between rows cn i (Ȳi Ȳ ) 2 r 1 σ 2 + cn r 1 i α 2 i between columns rn j (Ȳ j Ȳ ) 2 c 1 σ 2 + rn c 1 j β 2 j interaction error n i j (Ȳij Ȳi Ȳ j + Ȳ ) 2 (r 1) (c 1) i j k (Y ijk Ȳij ) 2 r c (n 1) σ 2 + σ 2 n (r 1) (c 1) i j γ 2 ij This is for fixed effects, and equal number of observations per cell!

38 Example (continued) source SS df MS F p-value sex treatment interaction error Interaction in a 2-way ANOVA model Let Y ijk be the k th item in the subgroup representing the i th group of factor A (r levels) and the j th group of factor B (c levels). We write Y ijk =µ + α i + β j + γ ij + ɛ ijk no interaction positive interaction negative interaction N A B A+B N A B A+B N A B A+B

39 Expected mean squares source fixed effects random effects mixed effects between rows σ 2 + cn r 1 i α 2 i σ 2 + n σ 2 R C + cnσ2 R σ 2 + n σ 2 R C + cn r 1 i α 2 i between columns σ 2 + rn c 1 interaction σ 2 + βj 2 σ 2 + n σr C 2 + rnσ2 C σ 2 + rnσc 2 n (r 1)(c 1) j γij 2 σ 2 + n σr C 2 σ 2 + n σr C 2 i j error σ 2 σ 2 σ 2 Two-way ANOVA without replicates Physician Concentration A B C

40 ANOVA table source df SS MS physician concentration interaction total 20 We have 21 observations. That means we have no degrees of freedom left to estimate an error! Expected mean squares In general, we have: source fixed effects random effects mixed effects between rows σ 2 + cn r 1 i α 2 i σ 2 + n σ 2 R C + cnσ2 R σ 2 + n σ 2 R C + cn r 1 i α 2 i between columns σ 2 + rn c 1 interaction σ 2 + βj 2 σ 2 + n σr C 2 + rnσ2 C σ 2 + rnσc 2 n (r 1)(c 1) j γij 2 σ 2 + n σr C 2 σ 2 + n σr C 2 i j error σ 2 σ 2 σ 2

41 Expected mean squares If n=1 and there is no interaction in truth, we have: source fixed effects random effects mixed effects between rows σ 2 + c r 1 between columns σ 2 + r c 1 i α 2 i σ 2 + c σ 2 R σ 2 + c r 1 βj 2 σ 2 + r σc 2 σ 2 + r σc 2 j i α 2 i error σ 2 σ 2 σ 2 Expected mean squares If n=1 but there is an interaction, we have: source fixed effects random effects mixed effects between rows σ 2 + c r 1 between columns σ 2 + r c 1 interaction σ 2 + i α 2 i σ 2 + σ 2 R C + c σ2 R σ 2 + σ 2 R C + c r 1 βj 2 σ 2 + σr C 2 + r σ2 C σ 2 + r σc 2 1 (r 1)(c 1) j γij 2 σ 2 + σr C 2 σ 2 + σr C 2 i j i α 2 i error σ 2 σ 2 σ 2

ANOVA Multiple Comparisons

ANOVA Multiple Comparisons ANOVA Multiple Comparisons Multiple comparisons When we carry out an ANOVA on k treatments, we test H 0 : µ 1 = =µ k versus H a : H 0 is false Assume we reject the null hypothesis, i.e. we have some evidence

More information

Multiple comparisons - subsequent inferences for two-way ANOVA

Multiple comparisons - subsequent inferences for two-way ANOVA 1 Multiple comparisons - subsequent inferences for two-way ANOVA the kinds of inferences to be made after the F tests of a two-way ANOVA depend on the results if none of the F tests lead to rejection of

More information

Chapter 12. Analysis of variance

Chapter 12. Analysis of variance Serik Sagitov, Chalmers and GU, January 9, 016 Chapter 1. Analysis of variance Chapter 11: I = samples independent samples paired samples Chapter 1: I 3 samples of equal size J one-way layout two-way layout

More information

Food consumption of rats. Two-way vs one-way vs nested ANOVA

Food consumption of rats. Two-way vs one-way vs nested ANOVA Food consumption of rats Lard Gender Fresh Rancid 709 592 Male 679 538 699 476 657 508 Female 594 505 677 539 Two-way vs one-way vs nested NOV T 1 2 * * M * * * * * * F * * * * T M1 M2 F1 F2 * * * * *

More information

Lec 1: An Introduction to ANOVA

Lec 1: An Introduction to ANOVA Ying Li Stockholm University October 31, 2011 Three end-aisle displays Which is the best? Design of the Experiment Identify the stores of the similar size and type. The displays are randomly assigned to

More information

I i=1 1 I(J 1) j=1 (Y ij Ȳi ) 2. j=1 (Y j Ȳ )2 ] = 2n( is the two-sample t-test statistic.

I i=1 1 I(J 1) j=1 (Y ij Ȳi ) 2. j=1 (Y j Ȳ )2 ] = 2n( is the two-sample t-test statistic. Serik Sagitov, Chalmers and GU, February, 08 Solutions chapter Matlab commands: x = data matrix boxplot(x) anova(x) anova(x) Problem.3 Consider one-way ANOVA test statistic For I = and = n, put F = MS

More information

1 One-way Analysis of Variance

1 One-way Analysis of Variance 1 One-way Analysis of Variance Suppose that a random sample of q individuals receives treatment T i, i = 1,,... p. Let Y ij be the response from the jth individual to be treated with the ith treatment

More information

QUEEN MARY, UNIVERSITY OF LONDON

QUEEN MARY, UNIVERSITY OF LONDON QUEEN MARY, UNIVERSITY OF LONDON MTH634 Statistical Modelling II Solutions to Exercise Sheet 4 Octobe07. We can write (y i. y.. ) (yi. y i.y.. +y.. ) yi. y.. S T. ( Ti T i G n Ti G n y i. +y.. ) G n T

More information

Factorial designs. Experiments

Factorial designs. Experiments Chapter 5: Factorial designs Petter Mostad mostad@chalmers.se Experiments Actively making changes and observing the result, to find causal relationships. Many types of experimental plans Measuring response

More information

Analysis of Variance (ANOVA) Cancer Research UK 10 th of May 2018 D.-L. Couturier / R. Nicholls / M. Fernandes

Analysis of Variance (ANOVA) Cancer Research UK 10 th of May 2018 D.-L. Couturier / R. Nicholls / M. Fernandes Analysis of Variance (ANOVA) Cancer Research UK 10 th of May 2018 D.-L. Couturier / R. Nicholls / M. Fernandes 2 Quick review: Normal distribution Y N(µ, σ 2 ), f Y (y) = 1 2πσ 2 (y µ)2 e 2σ 2 E[Y ] =

More information

Analysis of variance (ANOVA) Comparing the means of more than two groups

Analysis of variance (ANOVA) Comparing the means of more than two groups Analysis of variance (ANOVA) Comparing the means of more than two groups Example: Cost of mating in male fruit flies Drosophila Treatments: place males with and without unmated (virgin) females Five treatments

More information

Summary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)

Summary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1) Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ

More information

4/6/16. Non-parametric Test. Overview. Stephen Opiyo. Distinguish Parametric and Nonparametric Test Procedures

4/6/16. Non-parametric Test. Overview. Stephen Opiyo. Distinguish Parametric and Nonparametric Test Procedures Non-parametric Test Stephen Opiyo Overview Distinguish Parametric and Nonparametric Test Procedures Explain commonly used Nonparametric Test Procedures Perform Hypothesis Tests Using Nonparametric Procedures

More information

Lec 5: Factorial Experiment

Lec 5: Factorial Experiment November 21, 2011 Example Study of the battery life vs the factors temperatures and types of material. A: Types of material, 3 levels. B: Temperatures, 3 levels. Example Study of the battery life vs the

More information

2 Hand-out 2. Dr. M. P. M. M. M c Loughlin Revised 2018

2 Hand-out 2. Dr. M. P. M. M. M c Loughlin Revised 2018 Math 403 - P. & S. III - Dr. McLoughlin - 1 2018 2 Hand-out 2 Dr. M. P. M. M. M c Loughlin Revised 2018 3. Fundamentals 3.1. Preliminaries. Suppose we can produce a random sample of weights of 10 year-olds

More information

Chapter Seven: Multi-Sample Methods 1/52

Chapter Seven: Multi-Sample Methods 1/52 Chapter Seven: Multi-Sample Methods 1/52 7.1 Introduction 2/52 Introduction The independent samples t test and the independent samples Z test for a difference between proportions are designed to analyze

More information

22s:152 Applied Linear Regression. Take random samples from each of m populations.

22s:152 Applied Linear Regression. Take random samples from each of m populations. 22s:152 Applied Linear Regression Chapter 8: ANOVA NOTE: We will meet in the lab on Monday October 10. One-way ANOVA Focuses on testing for differences among group means. Take random samples from each

More information

Unit 12: Analysis of Single Factor Experiments

Unit 12: Analysis of Single Factor Experiments Unit 12: Analysis of Single Factor Experiments Statistics 571: Statistical Methods Ramón V. León 7/16/2004 Unit 12 - Stat 571 - Ramón V. León 1 Introduction Chapter 8: How to compare two treatments. Chapter

More information

Dr. Junchao Xia Center of Biophysics and Computational Biology. Fall /8/2016 1/38

Dr. Junchao Xia Center of Biophysics and Computational Biology. Fall /8/2016 1/38 BIO5312 Biostatistics Lecture 11: Multisample Hypothesis Testing II Dr. Junchao Xia Center of Biophysics and Computational Biology Fall 2016 11/8/2016 1/38 Outline In this lecture, we will continue to

More information

STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis

STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis STAT 135 Lab 9 Multiple Testing, One-Way ANOVA and Kruskal-Wallis Rebecca Barter April 6, 2015 Multiple Testing Multiple Testing Recall that when we were doing two sample t-tests, we were testing the equality

More information

Formal Statement of Simple Linear Regression Model

Formal Statement of Simple Linear Regression Model Formal Statement of Simple Linear Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters X i is a known constant, the value of the predictor

More information

Introduction. Chapter 8

Introduction. Chapter 8 Chapter 8 Introduction In general, a researcher wants to compare one treatment against another. The analysis of variance (ANOVA) is a general test for comparing treatment means. When the null hypothesis

More information

The legacy of Sir Ronald A. Fisher. Fisher s three fundamental principles: local control, replication, and randomization.

The legacy of Sir Ronald A. Fisher. Fisher s three fundamental principles: local control, replication, and randomization. 1 Chapter 1: Research Design Principles The legacy of Sir Ronald A. Fisher. Fisher s three fundamental principles: local control, replication, and randomization. 2 Chapter 2: Completely Randomized Design

More information

Outline Topic 21 - Two Factor ANOVA

Outline Topic 21 - Two Factor ANOVA Outline Topic 21 - Two Factor ANOVA Data Model Parameter Estimates - Fall 2013 Equal Sample Size One replicate per cell Unequal Sample size Topic 21 2 Overview Now have two factors (A and B) Suppose each

More information

http://www.statsoft.it/out.php?loc=http://www.statsoft.com/textbook/ Group comparison test for independent samples The purpose of the Analysis of Variance (ANOVA) is to test for significant differences

More information

PSY 307 Statistics for the Behavioral Sciences. Chapter 20 Tests for Ranked Data, Choosing Statistical Tests

PSY 307 Statistics for the Behavioral Sciences. Chapter 20 Tests for Ranked Data, Choosing Statistical Tests PSY 307 Statistics for the Behavioral Sciences Chapter 20 Tests for Ranked Data, Choosing Statistical Tests What To Do with Non-normal Distributions Tranformations (pg 382): The shape of the distribution

More information

G. Nested Designs. 1 Introduction. 2 Two-Way Nested Designs (Balanced Cases) 1.1 Definition (Nested Factors) 1.2 Notation. 1.3 Example. 2.

G. Nested Designs. 1 Introduction. 2 Two-Way Nested Designs (Balanced Cases) 1.1 Definition (Nested Factors) 1.2 Notation. 1.3 Example. 2. G. Nested Designs 1 Introduction. 1.1 Definition (Nested Factors) When each level of one factor B is associated with one and only one level of another factor A, we say that B is nested within factor A.

More information

Contents. TAMS38 - Lecture 6 Factorial design, Latin Square Design. Lecturer: Zhenxia Liu. Factorial design 3. Complete three factor design 4

Contents. TAMS38 - Lecture 6 Factorial design, Latin Square Design. Lecturer: Zhenxia Liu. Factorial design 3. Complete three factor design 4 Contents Factorial design TAMS38 - Lecture 6 Factorial design, Latin Square Design Lecturer: Zhenxia Liu Department of Mathematics - Mathematical Statistics 28 November, 2017 Complete three factor design

More information

22s:152 Applied Linear Regression. There are a couple commonly used models for a one-way ANOVA with m groups. Chapter 8: ANOVA

22s:152 Applied Linear Regression. There are a couple commonly used models for a one-way ANOVA with m groups. Chapter 8: ANOVA 22s:152 Applied Linear Regression Chapter 8: ANOVA NOTE: We will meet in the lab on Monday October 10. One-way ANOVA Focuses on testing for differences among group means. Take random samples from each

More information

Section 4.6 Simple Linear Regression

Section 4.6 Simple Linear Regression Section 4.6 Simple Linear Regression Objectives ˆ Basic philosophy of SLR and the regression assumptions ˆ Point & interval estimation of the model parameters, and how to make predictions ˆ Point and interval

More information

CHI SQUARE ANALYSIS 8/18/2011 HYPOTHESIS TESTS SO FAR PARAMETRIC VS. NON-PARAMETRIC

CHI SQUARE ANALYSIS 8/18/2011 HYPOTHESIS TESTS SO FAR PARAMETRIC VS. NON-PARAMETRIC CHI SQUARE ANALYSIS I N T R O D U C T I O N T O N O N - P A R A M E T R I C A N A L Y S E S HYPOTHESIS TESTS SO FAR We ve discussed One-sample t-test Dependent Sample t-tests Independent Samples t-tests

More information

Cuckoo Birds. Analysis of Variance. Display of Cuckoo Bird Egg Lengths

Cuckoo Birds. Analysis of Variance. Display of Cuckoo Bird Egg Lengths Cuckoo Birds Analysis of Variance Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison Statistics 371 29th November 2005 Cuckoo birds have a behavior in which they lay their

More information

Solution to Final Exam

Solution to Final Exam Stat 660 Solution to Final Exam. (5 points) A large pharmaceutical company is interested in testing the uniformity (a continuous measurement that can be taken by a measurement instrument) of their film-coated

More information

Stat 217 Final Exam. Name: May 1, 2002

Stat 217 Final Exam. Name: May 1, 2002 Stat 217 Final Exam Name: May 1, 2002 Problem 1. Three brands of batteries are under study. It is suspected that the lives (in weeks) of the three brands are different. Five batteries of each brand are

More information

Comparing the means of more than two groups

Comparing the means of more than two groups Comparing the means of more than two groups Chapter 15 Analysis of variance (ANOVA) Like a t-test, but can compare more than two groups Asks whether any of two or more means is different from any other.

More information

3. Design Experiments and Variance Analysis

3. Design Experiments and Variance Analysis 3. Design Experiments and Variance Analysis Isabel M. Rodrigues 1 / 46 3.1. Completely randomized experiment. Experimentation allows an investigator to find out what happens to the output variables when

More information

Analysis of Variance

Analysis of Variance Statistical Techniques II EXST7015 Analysis of Variance 15a_ANOVA_Introduction 1 Design The simplest model for Analysis of Variance (ANOVA) is the CRD, the Completely Randomized Design This model is also

More information

Multiple Sample Numerical Data

Multiple Sample Numerical Data Multiple Sample Numerical Data Analysis of Variance, Kruskal-Wallis test, Friedman test University of California, San Diego Instructor: Ery Arias-Castro http://math.ucsd.edu/~eariasca/teaching.html 1 /

More information

22s:152 Applied Linear Regression. Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA)

22s:152 Applied Linear Regression. Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA) 22s:152 Applied Linear Regression Chapter 8: 1-Way Analysis of Variance (ANOVA) 2-Way Analysis of Variance (ANOVA) We now consider an analysis with only categorical predictors (i.e. all predictors are

More information

Solutions to Final STAT 421, Fall 2008

Solutions to Final STAT 421, Fall 2008 Solutions to Final STAT 421, Fall 2008 Fritz Scholz 1. (8) Two treatments A and B were randomly assigned to 8 subjects (4 subjects to each treatment) with the following responses: 0, 1, 3, 6 and 5, 7,

More information

Multiple Pairwise Comparison Procedures in One-Way ANOVA with Fixed Effects Model

Multiple Pairwise Comparison Procedures in One-Way ANOVA with Fixed Effects Model Biostatistics 250 ANOVA Multiple Comparisons 1 ORIGIN 1 Multiple Pairwise Comparison Procedures in One-Way ANOVA with Fixed Effects Model When the omnibus F-Test for ANOVA rejects the null hypothesis that

More information

Lecture 14: ANOVA and the F-test

Lecture 14: ANOVA and the F-test Lecture 14: ANOVA and the F-test S. Massa, Department of Statistics, University of Oxford 3 February 2016 Example Consider a study of 983 individuals and examine the relationship between duration of breastfeeding

More information

DESAIN EKSPERIMEN Analysis of Variances (ANOVA) Semester Genap 2017/2018 Jurusan Teknik Industri Universitas Brawijaya

DESAIN EKSPERIMEN Analysis of Variances (ANOVA) Semester Genap 2017/2018 Jurusan Teknik Industri Universitas Brawijaya DESAIN EKSPERIMEN Analysis of Variances (ANOVA) Semester Jurusan Teknik Industri Universitas Brawijaya Outline Introduction The Analysis of Variance Models for the Data Post-ANOVA Comparison of Means Sample

More information

Research Methods II MICHAEL BERNSTEIN CS 376

Research Methods II MICHAEL BERNSTEIN CS 376 Research Methods II MICHAEL BERNSTEIN CS 376 Goal Understand and use statistical techniques common to HCI research 2 Last time How to plan an evaluation What is a statistical test? Chi-square t-test Paired

More information

One-way ANOVA (Single-Factor CRD)

One-way ANOVA (Single-Factor CRD) One-way ANOVA (Single-Factor CRD) STAT:5201 Week 3: Lecture 3 1 / 23 One-way ANOVA We have already described a completed randomized design (CRD) where treatments are randomly assigned to EUs. There is

More information

10 One-way analysis of variance (ANOVA)

10 One-way analysis of variance (ANOVA) 10 One-way analysis of variance (ANOVA) A factor is in an experiment; its values are. A one-way analysis of variance (ANOVA) tests H 0 : µ 1 = = µ I, where I is the for one factor, against H A : at least

More information

Chap The McGraw-Hill Companies, Inc. All rights reserved.

Chap The McGraw-Hill Companies, Inc. All rights reserved. 11 pter11 Chap Analysis of Variance Overview of ANOVA Multiple Comparisons Tests for Homogeneity of Variances Two-Factor ANOVA Without Replication General Linear Model Experimental Design: An Overview

More information

STAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test

STAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test STAT 135 Lab 10 Two-Way ANOVA, Randomized Block Design and Friedman s Test Rebecca Barter April 13, 2015 Let s now imagine a dataset for which our response variable, Y, may be influenced by two factors,

More information

Group comparison test for independent samples

Group comparison test for independent samples Group comparison test for independent samples The purpose of the Analysis of Variance (ANOVA) is to test for significant differences between means. Supposing that: samples come from normal populations

More information

The entire data set consists of n = 32 widgets, 8 of which were made from each of q = 4 different materials.

The entire data set consists of n = 32 widgets, 8 of which were made from each of q = 4 different materials. One-Way ANOVA Summary The One-Way ANOVA procedure is designed to construct a statistical model describing the impact of a single categorical factor X on a dependent variable Y. Tests are run to determine

More information

Confidence Intervals, Testing and ANOVA Summary

Confidence Intervals, Testing and ANOVA Summary Confidence Intervals, Testing and ANOVA Summary 1 One Sample Tests 1.1 One Sample z test: Mean (σ known) Let X 1,, X n a r.s. from N(µ, σ) or n > 30. Let The test statistic is H 0 : µ = µ 0. z = x µ 0

More information

Lecture 6: Single-classification multivariate ANOVA (k-group( MANOVA)

Lecture 6: Single-classification multivariate ANOVA (k-group( MANOVA) Lecture 6: Single-classification multivariate ANOVA (k-group( MANOVA) Rationale and MANOVA test statistics underlying principles MANOVA assumptions Univariate ANOVA Planned and unplanned Multivariate ANOVA

More information

Introduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test

Introduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test Introduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test la Contents The two sample t-test generalizes into Analysis of Variance. In analysis of variance ANOVA the population consists

More information

Unit 7: Random Effects, Subsampling, Nested and Crossed Factor Designs

Unit 7: Random Effects, Subsampling, Nested and Crossed Factor Designs Unit 7: Random Effects, Subsampling, Nested and Crossed Factor Designs STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Understand how to interpret a random effect Know the different

More information

MATH Notebook 3 Spring 2018

MATH Notebook 3 Spring 2018 MATH448001 Notebook 3 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2010 2018 by Jenny A. Baglivo. All Rights Reserved. 3 MATH448001 Notebook 3 3 3.1 One Way Layout........................................

More information

Chapter 11. Analysis of Variance (One-Way)

Chapter 11. Analysis of Variance (One-Way) Chapter 11 Analysis of Variance (One-Way) We now develop a statistical procedure for comparing the means of two or more groups, known as analysis of variance or ANOVA. These groups might be the result

More information

IEOR165 Discussion Week 12

IEOR165 Discussion Week 12 IEOR165 Discussion Week 12 Sheng Liu University of California, Berkeley Apr 15, 2016 Outline 1 Type I errors & Type II errors 2 Multiple Testing 3 ANOVA IEOR165 Discussion Sheng Liu 2 Type I errors & Type

More information

BIO 682 Nonparametric Statistics Spring 2010

BIO 682 Nonparametric Statistics Spring 2010 BIO 682 Nonparametric Statistics Spring 2010 Steve Shuster http://www4.nau.edu/shustercourses/bio682/index.htm Lecture 8 Example: Sign Test 1. The number of warning cries delivered against intruders by

More information

BIOL Biometry LAB 6 - SINGLE FACTOR ANOVA and MULTIPLE COMPARISON PROCEDURES

BIOL Biometry LAB 6 - SINGLE FACTOR ANOVA and MULTIPLE COMPARISON PROCEDURES BIOL 458 - Biometry LAB 6 - SINGLE FACTOR ANOVA and MULTIPLE COMPARISON PROCEDURES PART 1: INTRODUCTION TO ANOVA Purpose of ANOVA Analysis of Variance (ANOVA) is an extremely useful statistical method

More information

Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data

Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data 1999 Prentice-Hall, Inc. Chap. 10-1 Chapter Topics The Completely Randomized Model: One-Factor

More information

Linear Combinations. Comparison of treatment means. Bruce A Craig. Department of Statistics Purdue University. STAT 514 Topic 6 1

Linear Combinations. Comparison of treatment means. Bruce A Craig. Department of Statistics Purdue University. STAT 514 Topic 6 1 Linear Combinations Comparison of treatment means Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 6 1 Linear Combinations of Means y ij = µ + τ i + ǫ ij = µ i + ǫ ij Often study

More information

An Analysis of College Algebra Exam Scores December 14, James D Jones Math Section 01

An Analysis of College Algebra Exam Scores December 14, James D Jones Math Section 01 An Analysis of College Algebra Exam s December, 000 James D Jones Math - Section 0 An Analysis of College Algebra Exam s Introduction Students often complain about a test being too difficult. Are there

More information

Analysis of Variance. Read Chapter 14 and Sections to review one-way ANOVA.

Analysis of Variance. Read Chapter 14 and Sections to review one-way ANOVA. Analysis of Variance Read Chapter 14 and Sections 15.1-15.2 to review one-way ANOVA. Design of an experiment the process of planning an experiment to insure that an appropriate analysis is possible. Some

More information

Introduction to Analysis of Variance (ANOVA) Part 2

Introduction to Analysis of Variance (ANOVA) Part 2 Introduction to Analysis of Variance (ANOVA) Part 2 Single factor Serpulid recruitment and biofilms Effect of biofilm type on number of recruiting serpulid worms in Port Phillip Bay Response variable:

More information

Stat 6640 Solution to Midterm #2

Stat 6640 Solution to Midterm #2 Stat 6640 Solution to Midterm #2 1. A study was conducted to examine how three statistical software packages used in a statistical course affect the statistical competence a student achieves. At the end

More information

Ch. 5 Two-way ANOVA: Fixed effect model Equal sample sizes

Ch. 5 Two-way ANOVA: Fixed effect model Equal sample sizes Ch. 5 Two-way ANOVA: Fixed effect model Equal sample sizes 1 Assumptions and models There are two factors, factors A and B, that are of interest. Factor A is studied at a levels, and factor B at b levels;

More information

STAT 263/363: Experimental Design Winter 2016/17. Lecture 1 January 9. Why perform Design of Experiments (DOE)? There are at least two reasons:

STAT 263/363: Experimental Design Winter 2016/17. Lecture 1 January 9. Why perform Design of Experiments (DOE)? There are at least two reasons: STAT 263/363: Experimental Design Winter 206/7 Lecture January 9 Lecturer: Minyong Lee Scribe: Zachary del Rosario. Design of Experiments Why perform Design of Experiments (DOE)? There are at least two

More information

STAT 430 (Fall 2017): Tutorial 8

STAT 430 (Fall 2017): Tutorial 8 STAT 430 (Fall 2017): Tutorial 8 Balanced Incomplete Block Design Luyao Lin November 7th/9th, 2017 Department Statistics and Actuarial Science, Simon Fraser University Block Design Complete Random Complete

More information

Central Limit Theorem ( 5.3)

Central Limit Theorem ( 5.3) Central Limit Theorem ( 5.3) Let X 1, X 2,... be a sequence of independent random variables, each having n mean µ and variance σ 2. Then the distribution of the partial sum S n = X i i=1 becomes approximately

More information

Ch 3: Multiple Linear Regression

Ch 3: Multiple Linear Regression Ch 3: Multiple Linear Regression 1. Multiple Linear Regression Model Multiple regression model has more than one regressor. For example, we have one response variable and two regressor variables: 1. delivery

More information

More about Single Factor Experiments

More about Single Factor Experiments More about Single Factor Experiments 1 2 3 0 / 23 1 2 3 1 / 23 Parameter estimation Effect Model (1): Y ij = µ + A i + ɛ ij, Ji A i = 0 Estimation: µ + A i = y i. ˆµ = y..  i = y i. y.. Effect Modell

More information

Analysis of Variance

Analysis of Variance Analysis of Variance Math 36b May 7, 2009 Contents 2 ANOVA: Analysis of Variance 16 2.1 Basic ANOVA........................... 16 2.1.1 the model......................... 17 2.1.2 treatment sum of squares.................

More information

Parametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami

Parametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami Parametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami Parametric Assumptions The observations must be independent. Dependent variable should be continuous

More information

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă

HYPOTHESIS TESTING II TESTS ON MEANS. Sorana D. Bolboacă HYPOTHESIS TESTING II TESTS ON MEANS Sorana D. Bolboacă OBJECTIVES Significance value vs p value Parametric vs non parametric tests Tests on means: 1 Dec 14 2 SIGNIFICANCE LEVEL VS. p VALUE Materials and

More information

Group comparison test for independent samples

Group comparison test for independent samples Group comparison test for independent samples Samples come from normal populations with possibly different means but a common variance Two independent samples: z or t test on difference between means Three,

More information

Comparison of Two Samples

Comparison of Two Samples 2 Comparison of Two Samples 2.1 Introduction Problems of comparing two samples arise frequently in medicine, sociology, agriculture, engineering, and marketing. The data may have been generated by observation

More information

Lecture 9: Factorial Design Montgomery: chapter 5

Lecture 9: Factorial Design Montgomery: chapter 5 Lecture 9: Factorial Design Montgomery: chapter 5 Page 1 Examples Example I. Two factors (A, B) each with two levels (, +) Page 2 Three Data for Example I Ex.I-Data 1 A B + + 27,33 51,51 18,22 39,41 EX.I-Data

More information

STAT 5200 Handout #7a Contrasts & Post hoc Means Comparisons (Ch. 4-5)

STAT 5200 Handout #7a Contrasts & Post hoc Means Comparisons (Ch. 4-5) STAT 5200 Handout #7a Contrasts & Post hoc Means Comparisons Ch. 4-5) Recall CRD means and effects models: Y ij = µ i + ϵ ij = µ + α i + ϵ ij i = 1,..., g ; j = 1,..., n ; ϵ ij s iid N0, σ 2 ) If we reject

More information

Statistiek II. John Nerbonne using reworkings by Hartmut Fitz and Wilbert Heeringa. February 13, Dept of Information Science

Statistiek II. John Nerbonne using reworkings by Hartmut Fitz and Wilbert Heeringa. February 13, Dept of Information Science Statistiek II John Nerbonne using reworkings by Hartmut Fitz and Wilbert Heeringa Dept of Information Science j.nerbonne@rug.nl February 13, 2014 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated

More information

Tentative solutions TMA4255 Applied Statistics 16 May, 2015

Tentative solutions TMA4255 Applied Statistics 16 May, 2015 Norwegian University of Science and Technology Department of Mathematical Sciences Page of 9 Tentative solutions TMA455 Applied Statistics 6 May, 05 Problem Manufacturer of fertilizers a) Are these independent

More information

Week 14 Comparing k(> 2) Populations

Week 14 Comparing k(> 2) Populations Week 14 Comparing k(> 2) Populations Week 14 Objectives Methods associated with testing for the equality of k(> 2) means or proportions are presented. Post-testing concepts and analysis are introduced.

More information

COMPARING SEVERAL MEANS: ANOVA

COMPARING SEVERAL MEANS: ANOVA LAST UPDATED: November 15, 2012 COMPARING SEVERAL MEANS: ANOVA Objectives 2 Basic principles of ANOVA Equations underlying one-way ANOVA Doing a one-way ANOVA in R Following up an ANOVA: Planned contrasts/comparisons

More information

Lec 3: Model Adequacy Checking

Lec 3: Model Adequacy Checking November 16, 2011 Model validation Model validation is a very important step in the model building procedure. (one of the most overlooked) A high R 2 value does not guarantee that the model fits the data

More information

Distribution-Free Procedures (Devore Chapter Fifteen)

Distribution-Free Procedures (Devore Chapter Fifteen) Distribution-Free Procedures (Devore Chapter Fifteen) MATH-5-01: Probability and Statistics II Spring 018 Contents 1 Nonparametric Hypothesis Tests 1 1.1 The Wilcoxon Rank Sum Test........... 1 1. Normal

More information

Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2

Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Fall, 2013 Page 1 Random Variable and Probability Distribution Discrete random variable Y : Finite possible values {y

More information

Ch 2: Simple Linear Regression

Ch 2: Simple Linear Regression Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component

More information

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE

THE ROYAL STATISTICAL SOCIETY HIGHER CERTIFICATE THE ROYAL STATISTICAL SOCIETY 004 EXAMINATIONS SOLUTIONS HIGHER CERTIFICATE PAPER II STATISTICAL METHODS The Society provides these solutions to assist candidates preparing for the examinations in future

More information

Lecture 7: Hypothesis Testing and ANOVA

Lecture 7: Hypothesis Testing and ANOVA Lecture 7: Hypothesis Testing and ANOVA Goals Overview of key elements of hypothesis testing Review of common one and two sample tests Introduction to ANOVA Hypothesis Testing The intent of hypothesis

More information

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007)

CHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007) FROM: PAGANO, R. R. (007) I. INTRODUCTION: DISTINCTION BETWEEN PARAMETRIC AND NON-PARAMETRIC TESTS Statistical inference tests are often classified as to whether they are parametric or nonparametric Parameter

More information

Design & Analysis of Experiments 7E 2009 Montgomery

Design & Analysis of Experiments 7E 2009 Montgomery Chapter 5 1 Introduction to Factorial Design Study the effects of 2 or more factors All possible combinations of factor levels are investigated For example, if there are a levels of factor A and b levels

More information

CHAPTER 4 Analysis of Variance. One-way ANOVA Two-way ANOVA i) Two way ANOVA without replication ii) Two way ANOVA with replication

CHAPTER 4 Analysis of Variance. One-way ANOVA Two-way ANOVA i) Two way ANOVA without replication ii) Two way ANOVA with replication CHAPTER 4 Analysis of Variance One-way ANOVA Two-way ANOVA i) Two way ANOVA without replication ii) Two way ANOVA with replication 1 Introduction In this chapter, expand the idea of hypothesis tests. We

More information

Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) Analysis of Variance (ANOVA) Much of statistical inference centers around the ability to distinguish between two or more groups in terms of some underlying response variable y. Sometimes, there are but

More information

Factorial and Unbalanced Analysis of Variance

Factorial and Unbalanced Analysis of Variance Factorial and Unbalanced Analysis of Variance Nathaniel E. Helwig Assistant Professor of Psychology and Statistics University of Minnesota (Twin Cities) Updated 04-Jan-2017 Nathaniel E. Helwig (U of Minnesota)

More information

Lecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series

Lecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 13 Nonparametric Statistics 13-1 Overview 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks

More information

SEVERAL μs AND MEDIANS: MORE ISSUES. Business Statistics

SEVERAL μs AND MEDIANS: MORE ISSUES. Business Statistics SEVERAL μs AND MEDIANS: MORE ISSUES Business Statistics CONTENTS Post-hoc analysis ANOVA for 2 groups The equal variances assumption The Kruskal-Wallis test Old exam question Further study POST-HOC ANALYSIS

More information

Lecture Slides. Section 13-1 Overview. Elementary Statistics Tenth Edition. Chapter 13 Nonparametric Statistics. by Mario F.

Lecture Slides. Section 13-1 Overview. Elementary Statistics Tenth Edition. Chapter 13 Nonparametric Statistics. by Mario F. Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 13 Nonparametric Statistics 13-1 Overview 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks

More information

44.2. Two-Way Analysis of Variance. Introduction. Prerequisites. Learning Outcomes

44.2. Two-Way Analysis of Variance. Introduction. Prerequisites. Learning Outcomes Two-Way Analysis of Variance 44 Introduction In the one-way analysis of variance (Section 441) we consider the effect of one factor on the values taken by a variable Very often, in engineering investigations,

More information

Lecture 10. Factorial experiments (2-way ANOVA etc)

Lecture 10. Factorial experiments (2-way ANOVA etc) Lecture 10. Factorial experiments (2-way ANOVA etc) Jesper Rydén Matematiska institutionen, Uppsala universitet jesper@math.uu.se Regression and Analysis of Variance autumn 2014 A factorial experiment

More information

Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing

Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire

More information

Basic Business Statistics, 10/e

Basic Business Statistics, 10/e Chapter 1 1-1 Basic Business Statistics 11 th Edition Chapter 1 Chi-Square Tests and Nonparametric Tests Basic Business Statistics, 11e 009 Prentice-Hall, Inc. Chap 1-1 Learning Objectives In this chapter,

More information