Example Problem: Factorial ANOVA

Ψ320 Ainsworth 10 years old 15 years old Age of Child 5 years old Example problem: Factorial ANOVA A researcher is interested in what affects the amount of candy that is given out during Halloween. She randomly selects 15 5-year-olds, 15 10-year-olds and 15 15-year-olds and randomly assigns them to wear either a funny, cute or scary costume. Each participant is sent to the same randomly selected house in a middle income neighborhood to trick-or-treat and the number of pieces of candy was recorded for each participant. Type of Halloween Costume Funny Cute Scary Marginal Means 12 11 16 Funny = 10.867 Cute = 6.067 Scary = 13.467 12 8 18 5 = 12.933 10 = 10.133 15 = 7.333 14 8 17 15 9 12 Marginal Sums 16 9 17 Funny = 163 Cute = 91 Scary = 202 14 6 11 5 = 194 10 = 152 15 = 110 14 2 10 15 7 12 Grand Mean = 10.13333 13 7 13 Total = 456 11 5 12 ΣY2 = 5514 5 4 14 5 6 12 5 3 13 4 1 9 8 5 16 Cell Means Funny Cute Scary 13.8 9.0 16.0 5 13.4 5.4 11.6 10 5.4 3.8 12.8 15 Cell Sums Funny Cute Scary 69 45 80 5 67 27 58 10 27 19 64 15 Cell Standard Deviations Funny Cute Scary 1.789 1.225 2.345 5 1.517 2.074 1.140 10 1.517 1.924 2.588 15 On the next page, perform a Factorial ANOVA and show all 7 hypothesis testing steps. Ψ320 Ainsworth 1. State Null Hypothesis: i. h0 : µfunny ___ µcute ___ µscary (Main effect for Costume) ii. h0 : µage5 ___ µage10 ___ µage15 (Main effect for Age) h0 : µage5funny ___ µ age5cute ___ µage5scary ___ µage10funny ___ µ age10cute ___ µage15scary iii. µage15funny ___ µ age15cute ___ µage15scary (Interaction Effect) 2. Alternative Hypothesis: i. h1 : At least 2 costume µ s are different ii. h1 : At least 2 Age µ s are different iii. h1 : At least 2 Cell µ s are different 3. Decide on α (usually .05): α = ____ 4. Decide on type of test (distribution; z, t, F, etc.) Questions to ask: a. How many groups do you have? i. Only 2 than you can use a t-test. ii. Are the groups arranged within multiple independent variables? iii. More than 2 groups/IVs → ANOVA → F distribution b. Can we treat the scores as independent (e.g. they are NOT from the same person, they are NOT matched subjects, they are NOT related subjects, etc.)? If Yes, then continue with the between groups ANOVA If No, STOP you may need to perform a repeated measures ANOVA c. Can we assume a normally distributed sampling distribution? In other words, do we have 20+ degrees of freedom for the WG source of variance? If yes, then continue. If no, do not continue the test cannot be performed. d. Do the groups have homogenous variances? FMAX = 2 sLargest 2 Smallest s = _____ = _______ , if this value is smaller than 3 proceed. _____ 5. Find critical value & state decision rule(s) a. For Fcv you need both df Effect = #groups(effect) – 1 and dfWG = N - #cells. Table D.3 Fcv ( df effect , dfWG ), if Fo > Fcv reject the null hypothesis b. So there are 3 costume groups therefore df Costume = ____ – 1 = ____ and dfWG = ____ - 1 = 36. Table D.3 Fcv (___,___) = ______, if Fo > _______ reject the null hypothesis Ψ320 Ainsworth c. So there are 3 age groups therefore df Age = ____ – 1 = ____ and dfWG = ____ - 1 = 36. Table D.3 Fcv (___,___) = ______, if Fo > _______ reject the null hypothesis d. So there are 9 cells for the interaction therefore df Interaction ( Costume* Age ) = ____ – 1 = ____ and dfWG = ____ - 1 = 36. Table D.3 Fcv (___,___) = ______, if Fo > _______ reject the null hypothesis 6. Calculate test • You can use the Deviation Approach but it will take you a long time, so for the sake of our collective sanities let’s just focus on the computational approach (but the deviation approach should achieve the same results SScostume ∑ ( ∑ A) = 2 bn 2 ( ___ ) T 2 1632 + ___ 2 + ___ 2 _____ _____ − = − = − = abn ___ ( __ )( __ ) ( __ )( __ )( __ ) ___ = ______ − ______ = ______ SS Age ∑ (∑ B) = an 2 − T2 ___ 2 + ___ 2 + 1102 ______ = − 4620.8 = − 4620.8 = abn ___ ( __ )( __ ) = ______ − 4620.8 = ______ 2 2 ∑ ( ∑ AB ) − ∑ ( ∑ A) − ∑ ( ∑ B ) = 2 T2 SS AB + = n bn an abn 692 + 452 + ___ 2 + ___ 2 + ___ 2 + ___ 2 + ___ 2 + ___ 2 + ___ 2 − 5043.6 − 4856 + 4620.8 = __ ______ − 5043.6 − 4856 + 4620.8 = ______ − 5043.6 − 4856 + 4620.8 = 112 __ SS S / AB = ∑ Y 2 ∑ ( ∑ AB ) − SSTotal = ∑ Y 2 − n 2 = 5514 − 5390.8 = _____ T2 = 5514 − 4620.8 = _____ abn Ψ320 Ainsworth Source Costume Age AB S/AB Total SS ____ ____ 112 ____ ____ df __ __ __ 36 MS F ____ ____ ____ ____ 28 ____ ____ 7. Apply Decision Rule(s) • Costume: Since, _______ (i.e. observed value) ____ (i.e. >, <) _______ (critical value), _________(i.e. DO or DO NOT) reject the null hypothesis. • Age: Since, _______ (i.e. observed value) ____ (i.e. >, <) _______ (critical value), _________(i.e. DO or DO NOT) reject the null hypothesis. • Interaction: Since, _______ (i.e. observed value) ____ (i.e. >, <) _______ (critical value), _________(i.e. DO or DO NOT) reject the null hypothesis.