1、ANalysis Of VAriance (ANOVA),Comparing 2 meansFrequently applied to experimental dataWhy not do multiple t-tests? If you want to testH0: m1 = m2 = m3Why not test: m1 = m2 m1 = m3 m2 = m3,For each test 95% probability to correctly fail to reject (accept?) null, when null is really true,0.953 = probab
2、ility of correctly failing to reject all 3 = 0.86,Probability of if incorrectly rejecting at least one of the (true) null hypotheses = 1 - 0.86 = 0.14As you increase the number of means compared, the probability of incorrectly rejecting a true null (type I error) increases towards oneSide note: poss
3、ible to correct (lower) if you need to do multiple tests (Bonferroni correction)- unusual,ANOVA: calculate ratios of different portions of variance of total dataset to determine if group means differ significantly from each otherCalculate F ratio, named after R.A. Fisher,1) Visualize data sets 2) Pa
4、rtition variance (SS & df) 3) Calculate F (tomorrow),0,1,2,3,4,5,6,7,8,0,10,20,30,Plot number,Yield (tonnes),Pictures first,3 fertilizers applied to 10 plots each (N=30), yield measuredHow much variability comes from fertilizers, how much from other factors?,Fert 1,Fert 2,Fert 3,Overall mean,What fa
5、ctors other than fertilizer (uncontrolled) may contribute to the variance in crop yield?How do you minimize uncontrolled factors contribution to variance when designing an experiment or survey study?If one wants to measure the effect of a factor in nature (most of ecology/geology), how can or should
6、 you minimize background variability between experimental units?,Thought Questions,Fertilizer (in this case) is termed the independent or predictor variable or explanatory variableCan have any number of levels, we have 3Can have more than one independent variable. We have 1, one way ANOVACrop yield
7、(in this case) is termed the dependent or response variableCan have more than one response variable. multivariate analysis (ex MANOVA). Class taught by J. Harrell,0,1,2,3,4,5,6,7,8,0,10,20,30,Plot number,Yield (tonnes),Pictures first,-calculate deviation of each point from mean -some + and some - -s
8、um to zero (remember definition of mean),Fert 1,Fert 2,Fert 3,Overall mean,Square all values,Sum the squared values,n-1,calculate mean SS a.k.a. variance,=,* why (n-1)? Because all deviations must sum to zero, therefore if you calculate n-1 deviations, you know what the final one must be. You do not
9、 actually have n independent pieces of information about the variance.,SS not useful for comparing between groups, it is always big when n is big. Using the mean SS (variance) allows you to compare among groups,Back to the question:How much variability in crop yield comes from fertilizers (what you
10、manipulated), how much from other factors (that you cannot control)?,Partitioning Variability,Calculate mean for each group, ie plots with fert1, fert2, and fert3 (3 group means),But first imagine a data set where,0,1,2,3,4,5,6,7,8,0,10,20,30,Plot number,Yield (tonnes),-Imagine case were the group (
11、treatment) means differ a lot, with little variation within a group-Group means explain most of the variability,Fert 1,Fert 2,Fert 3,Overall mean,Group means,0,1,2,3,4,5,6,7,8,0,10,20,30,Plot number,Yield (tonnes),Fert 1,Fert 2,Fert 3,Overall mean,Now. imagine case were the group (treatment) means a
12、re not distinct, with much variation within a group-Group means explain little of the variability -3 fertilizers did not affect yield differently,Group means,H0: mean yield fert1= mean yield fert2 = mean yield fert3Or Fertilizer type has no effect on crop yield,-calculating 3 measures of variability
13、,start by partitioning SS,Total SS =,Sum of squares of deviations of data around the grand (overall) mean (measure of total variability),Within group SS = (Error SS),Sum of squares of deviations of data around the separate group means (measure of variability among units given same treatment),Among g
14、roups SS =,Sum of squares of deviations of group means around the grand mean (measure of variability among units given different treatments),Unfortunate word usage,Total SS =,Sum of squares of deviations of data around the grand (overall) mean (measure of total variability),k = number experimental g
15、roups Xij = datum j in experimental group I Xbari = mean of group I Xbar = grand mean,k,i=1,ni,j=1,Xij - X,2,Total SS =,Total SS =,Sum of deviations of each datum from the grand mean, squared, summed across all k groups,Within group SS =,k,i=1,ni,j=1,Xij - Xi,2,Within group SS =,k = number experimen
16、tal groups Xij = datum j in experimental group I Xbari = mean of group I Xbar = grand mean,Within group SS =,Sum of squares of deviations of data around the separate group means (measure of variability among units given same treatment),Sum of deviations of each datum from its group mean, squared, su
17、mmed across all k groups,Among groups SS =,Sum of squares of deviations of group means around the grand mean (measure of variability among units given different treatments),k = number experimental groups Xij = datum j in experimental group I Xbari = mean of group I Xbar = grand mean,k,i=1,ni,Xi - X,
18、2,Among groups SS =,Among groups SS =,Sum of deviations of each group mean from the grand mean, squared,partitioning DF,Total df =,Within group df = (Error df),Among groups df =,Total number experimental units -1 In fertilizer experiment, n-1= 29,units in each group -1, summed for all groups In fert
19、ilizer experiment, (10-1)*3; 9*3=27,Number group means -1 In fertilizer experiment, 3-1=2,Unfortunate word usage,SS and df sum,Total SS = within groups SS + among groups SSTotal df = within groups df + among groups df,Mean squares,Combine information on SS and df,Total mean squares = total SS/ total
20、 dftotal variance of data set,Within group mean squares = within SS/ within dfvariance (per df) among units given same treatment,Among groups mean squares = among SS / among dfvariance (per df) among units given different treatments,Unfortunate word usage,Error MS,Tomorrow:the big Fexample calculati
21、ons,Mean squares,Combine information on SS and df,Total mean squares = total SS/ total dftotal variance of data set,Within group mean squares = within SS/ within dfvariance (per df) among units given same treatment,Among groups mean squares = among SS / among dfvariance (per df) among units given di
22、fferent treatments,Unfortunate word usage,Error MS,Among groups mean squares,Within group mean squares,F =, Back to the question: Does fitting the treatment mean explain a significant amount of variance?In our example. if fertilizer doesnt influence yield, then variation between plots with the same
23、fertilizer will be about the same as variation between plots given different fertilizers,Compare calculated F to critical value from table (B4),If calculated F as big or bigger than critical value, then reject H0But remember.H0: m1 = m2 = m3Need separate test (multiple comparison test) to tell which
24、 means differ from which,See handout,Remember Shape of t-distribution approaches normal curve as sample size gets very largeBut.F distribution is differentalways positive skewshape differs with df,Two types of ANOVA: fixed and random effects models,Among groups mean squares,Within group mean squares
25、,F =,Calculation of F as:,Assumes that the levels of the independent variable have been specifically chosen, as opposed to being randomly selected from a larger population of possible levels,ExsFixed: Test for differences in growth rates of three cultivars of roses. You want to decide which of the t
26、hree to plant. Random: Randomly select three cultivars of roses from a seed catalogue in order to test whether, in general, rose cultivars differ in growth rate Fixed: Test for differences in numbers of fast food meals consumed each month by students at UT, BG, and Ohio State in order to determine w
27、hich campus has healthier eating habitsRandom: Randomly select 3 college campuses and test whether the number of fast food meals per month differs among college campuses in general,In random effects ANOVA the denominator is not the within groups mean squaresProper denominator depends on nature of th
28、e question*Be aware that default output from most stats packages (eg, Excel, SAS) is fixed effect model,Assumptions of ANOVAAssumes that the variances of the k samples are similar (homogeneity of variance of homoscedastic) robust to violations of this assumption, especially when all ni are equalAssu
29、mes that the underlying populations are normally distributedalso robust to violations of this assumption,Model Formulae,Expression of the questions being asked,Does fertilizer affect yield?,yield = fertilizer,(word equation),response var,explanatory var,Right side can get more complicated,General Li
30、near Models,Linear models relating response and explanatory variables and encompassing ANOVA (& related tests) which have categorical explanatory variables and regression (& related tests) which have categorical explanatory variables,In SAS proc glm executes ANOVA, regression and other similar linea
31、r modelsOther procedures can also be used, glm is most general,data start;infile C:Documents and Settingscmayer3My DocumentsteachingBiostatisticsLecturesANOVA demo.csv dlm=, DSD;input plot fertilizer yield;options ls=80;proc print; data one; set start;proc glm;class fertilizer;model yield=fertilizer
32、; run;,The SAS System 512:53 Thursday, September 22, 2005The GLM ProcedureClass Level InformationClass Levels Valuesfertilizer 3 1 2 3Number of Observations Read 30Number of Observations Used 30The SAS System 612:53 Thursday, September 22, 2005The GLM Procedure Dependent Variable: yieldSum ofSource
33、DF Squares Mean Square F Value Pr FModel 2 10.82274667 5.41137333 5.70 0.0086Error 27 25.62215000 0.94896852Corrected Total 29 36.44489667R-Square Coeff Var Root MSE yeild Mean0.296962 20.97804 0.974150 4.643667Source DF Type I SS Mean Square F Value Pr Ffertilizer 2 10.82274667 5.41137333 5.70 0.0086Source DF Type III SS Mean Square F Value Pr Ffertilizer 2 10.82274667 5.41137333 5.70 0.0086,
