Unit 7 Analysis of Variance Practice Problems - 1 of 2 SOLUTIONS Stata

Transcription

1 Unt 7 Analyss of Varance Practce Problems - 1 of 2 SOLUTIONS Stata Before you begn: Download from the course webste: Stata Users anova_nfants.dta fshgrowth.dta Practce wth one way analyss of varance Exercses #1-6 Data set: anova_nfants.dta Zelazo et al. (1972) nvestgated the varablty n age at frst walkng n nfants. Study nfants were grouped nto four groups, accordng to renforcement of walkng and placement: (1) actve (2) passve (3) no exercse; and (4) 8 week control. Sample szes were 6 per group, for a total of n=24. For each nfant, study data ncluded group assgnment and age at frst walkng, n months. The followng are the data and consst of recorded values of age (months) by group: Actve Group Passve Group No-Exercse Group 8 Week Control Source: Zelazo et al (1972) Walkng n the newborn. Scence 176: Sol_anova_1 of 2 STATA.docx Page 1 of 13

2 2 1. State the analyss of varance model usng notaton µ and τ and σ as approprate. Defne all terms and constrants on the parameters Answer: 4 2 Y = µ + τ + ε, where ε ~N(0,σ ) and τ=0 å j j j e = 1, 2,... K ndexes method of renforcement group; K = number of groups = 4 j=1, 2,..., n =6 ndexes nfant wthn group; µ = populaton mean age at frst walkng, over all groups µ = mean age at frst walkng for nfants n group "" τ = [ µ - µ ] Y = observed age at frst walkng for the jth nfant n group "" j H: t =0, t =0, t =0, and t =0 O H : At least one t ¹ 0 A =1 2. By any means you lke, produce a sde by sde box plot showng the dstrbuton of age at frst walkng, separately for each of the 4 groups.. sort group. tabstat age, by(group) stat(n mean sd sem mn q max) Summary for varables: age by categores of: group group N mean sd se(mean) mn p25 p50 p75 max Total In these data, frst walkng occurs earler when nfants are renforced - Dstrbutons dffer markedly wth respect to varablty wth greatest seen among nfants n the passve group and smallest among nfants n the control group Sol_anova_1 of 2 STATA.docx Page 2 of 13

3 - Dstrbutons also dffer markedly n ther patterns of symmetry wth long rght tals n the actve and passve groups, long left tal n the no-exercse group, and symmetry n controls. set scheme s1color. label defne groupf 1 "Actve" 2 "Passve" 3 "No exercse" 4 "Control". label values group groupf. * No frlls graph. graph box age, over(group). * Same graph wth added aesthetcs.. graph box age, over(group, descendng) ntensty(50) box(1, bcolor(dknavy)) marker(1, msymbol(d) msze(medum) mcolor(dknavy)) ylabel(8(2)16, labsze(small)) yttle("month") ttle("age (months) at Frst Walkng, n=24") subttle("by Method of Renforcement") capton("exercse2.png", sze(vsmall)) NO Frlls Wth Aesthetcs Ex2_nofrlls.png exercse2.png - Plot confrms mpressons from the descrptve statstcs. Sol_anova_1 of 2 STATA.docx Page 3 of 13

4 3. By any means you lke, obtan the entres of the analyss of varance table for ths one way analyss of varance. Use your computer output (or excel work or hand calculatons or whatever) to complete the followng table: Sum of Squares SSQ Mean Square MSQ F-Statstc p-value Source df Between Groups Wthn Groups Total, corrected oneway age group, tabulate Summary of age group Mean Std. Dev. Freq Total Analyss of Varance Source SS df MS F Prob > F Between groups Wthn groups Total Bartlett's test for equal varances: ch2(3) = Prob>ch2 = Sol_anova_1 of 2 STATA.docx Page 4 of 13

5 4. Wrte a 2-5 sentence report of your descrpton and hypothess test fndngs usng language as approprate for a clent who s ntellgent but s not knowledgeable about statstcs. Include fgure and table as you thnk s approprate. In ths sample, the data suggest a trend towards earler age at frst walkng wth ncreasng renforcement and placement. The medan age at frst walkng s greatest among controls (12.35 months) and lowest among nfants n the actve group (10.13 months); see also the box plots. Tests of statstcal sgnfcance were lmted to the overall F test for group dfferences and ths dd not acheve statstcal sgnfcance (p-value =.10), possbly due to the small sample szes (6 n each group). Interestngly, examnaton of the data also suggests that the varablty n age at frst walkng dffered, dependng on the nterventon receved. The varablty was greater n the three nterventon groups ( actve, passve, no exercse ) compared to n the control group; ths was not statstcally sgnfcant however (p-value =.45). Further study, utlzng larger sample szes and addtonal hypothess tests to nvestgate trend are needed. 5. For the brave Usng approprately defned ndcator varables, perform a multvarable lnear regresson analyss of these same data! Use your computer output to complete the followng table: Source df Sum of Squares Mean Square Overall F due model (p) = 3 n SSR = Yˆ -Y = SSR/p = due error (resdual) (n-1-p) = 20 Total, corrected (n-1) = 23 å = 1 n å = 1 n ( ) 2 ( ˆ ) 2 SSE = Y -Y å( ) 2 SST = Y -Y = 1 = SSE/(n-1-p) =2.18 = Some of ths has already been done for you: I consdered the followng parameterzaton Y = age at frst walkng I_act = 0/1 ndcator of group assgnment to actve I_pass = 0/1 ndcator of group assgnment to passve I_noex = 0/1 ndcator of group assgnment to no exercse Thus, I used a reference cell codng approach wth 8 week control as my reference. I ft the followng multvarable lnear model of Y Y = b 0 + b 1 [I_act] + b 2 [I_pass] + b 3 [I_noex] + error Sol_anova_1 of 2 STATA.docx Page 5 of 13

6 . * I have already done ths. You do NOT need to reproduce these varable creatons.. generate I_actve=(group==1). generate I_pass=(group==2). generate I_noex=(group==3). regress age I_actve I_pass I_noex Source SS df MS Number of obs = F( 3, 20) = 2.40 Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = age Coef. Std. Err. t P> t [95% Conf. Interval] I_actve I_pass I_noex _cons The predcton equaton s thus: Ŷ = *I_act *I_pass *I_noex The two analyses n Stata match (hooray), thus confrmng that a multple lnear regresson model utlzng approprately defned ndcator varables s equvalent to an analyss of varance. Sol_anova_1 of 2 STATA.docx Page 6 of 13

7 6. For the brave: Usng your output from your two analyses (1 st -analyss of varance, 2 nd regresson), obtan the predcted mean of Y =age at frst walkng twce n two ways. Predcton Usng One Way Analyss of Varance Actve Passve No-Exercse Control Predcton Usng Multple Lnear Regresson ˆµ 1 = (β+β) ˆ ˆ 0 1 = = ˆµ 2 = (β+β) ˆ ˆ 0 2 = = ˆµ 3 = (β+β) ˆ ˆ 0 3 = = ˆµ = β ˆ = anova age group -- output omtted ---. adjust, by(group) Dependent varable: age Command: anova group xb Key: xb = Lnear Predcton Sol_anova_1 of 2 STATA.docx Page 7 of 13

8 Practce wth two-way factoral analyss of varance Exercses #7-12 Data set used: fshgrowth.dta Consder agan the fsh growth data on page 38 of Notes 7. Introducton to Analyss of Varance. Lght (lght) Water Temp (temp) Fsh Growth (growth) 1=low 1=cold =low 1=cold =low 2=lukewarm =low 2=lukewarm =low 3=warm =low 3=warm =hgh 1=cold =hgh 1=cold =hgh 2=lukewarm =hgh 2=lukewarm =hgh 3=warm =hgh 3=warm 6.92 Codng Manual: Varable growth lght temp Codng contnuous 1 = low 2 = hgh 1 = cold 2 = lukewarm 3 = warm Sol_anova_1 of 2 STATA.docx Page 8 of 13

9 2 7. State the analyss of varance model usng notaton µ, α, β, (αβ) and σ as approprate. Defne all terms and constrants on the parameters j j Answer: ( ) 2 Y = µ + α + β + αβ + ε, where ε ~N(0,σ ) jk j j jk jk wth = 1, 2 ndexng lght ( αβ ) j = 1, 2, 3 ndexng temperature k = 1, 2 ndexng ndvdual fsh under lght "" at water temperature "j" µ = populaton mean fsh growth, over all groups 2 å α = effect of lght level "", wth α =0 j j =1 3 å β = effect of water temperature "j", wth β=0 j=1 j = nteracton effect of the combnaton of "th" lght level and "jth" water temperature 2 3 å j å =1 j=1 wth (αβ) = 0 and (αβ) = 0. j 8. Create the followng new varables wth accompanyng defntons Varable Codng _hgh = 1 f (lght s 2) 0 otherwse _luke = 1 f (temp s 2) 0 otherwse _warm = 1 f (temp s 3) 0 otherwse h_luke = (_hgh) * (_luke) h_warm = (_hgh)*(_warm). generate _hgh=(lght==2). generate _luke=(temp==2). generate _warm=(temp==3). generate h_luke=_hgh*_luke. generate h_warm=_hgh*_warm Sol_anova_1 of 2 STATA.docx Page 9 of 13

10 9. Perform a two way analyss of varance so as to reproduce the followng table Source Df SSQ MSQ F p-value Due LIGHT Due TEMP Due Interacton Error Total (Corrected) anova growth lght temp lght#temp Number of obs = 12 R-squared = Root MSE = Adj R-squared = Source Partal SS df MS F Prob > F Model lght temp lght#temp Resdual Total Perform a regresson analyss to obtan the followng table and estmates Source Df SSQ MSQ F p-value Due Model Due Resdual Total (Corrected) Predctor beta Se(beta) T=beta/se p-value I_hgh I_luke I_warm H_luke H_warm Intercept Sol_anova_1 of 2 STATA.docx Page 10 of 13

11 . regress growth _hgh _luke _warm h_luke h_warm Source SS df MS Number of obs = F( 5, 6) = 5.74 Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = growth Coef. Std. Err. t P> t [95% Conf. Interval] _hgh _luke _warm h_luke h_warm _cons Usng the output from each of your anova and regresson analyses, complete the followng tables and notce that they are the same. Estmated Mean Growth, by Condtons of Lght and Temperature Anova Analyss Cold Lukewarm Warm Low lght Hgh lght Sol_anova_1 of 2 STATA.docx Page 11 of 13

12 . anova growth lght temp lght#temp ---- output omtted here --. adjust, by(lght temp) Number of obs = 12 R-squared = Root MSE = Adj R-squared = Source Partal SS df MS F Prob > F Model lght temp lght#temp Resdual Total Dependent varable: growth Command: anova temp lght 1=cold 2=lukewarm 3=warm =low =hgh Key: Lnear Predcton Sol_anova_1 of 2 STATA.docx Page 12 of 13

13 Estmated Mean Growth, by Condtons of Lght and Temperature Regresson Analyss Cold Lukewarm Warm Low lght Hgh lght regress growth _hgh _luke _warm h_luke h_warm ---- output omtted here --. predct predcted_growth. table lght temp, contents(mean predcted_growth) temp lght 1=cold 2=lukewarm 3=warm =low =hgh Compare your answer to #11 wth the observed means. Observed Mean Growth, by Condtons of Lght and Temperature Cold Lukewarm Warm Low lght Hgh lght They match! Sol_anova_1 of 2 STATA.docx Page 13 of 13