Lectures 15/16 ANOVA. ANOVA Tests. Analysis of Variance. >ANOVA stands for ANalysis Of VAriance >ANOVA allows us to:

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1 Lectures 5/6 Analysis of Variance ANOVA >ANOVA stands for ANalysis Of VAriance >ANOVA allows us to: Do multiple tests at one time more than two groups Test for multiple effects simultaneously more than one variable ANOVA Tests The types of ANONA we will look at are: >One Way ANOVA >Randomized block design ANOVA >Two-Factor >We will also see ANOVA in regression analysis 3

2 One-Way ANOVA >One-way ANOVA allows us to simultaneously test to determine if two or more population means are equal H O : µ = µ = µ 3 H A : At least two means differ 4 ANOVA assumptions >All populations are normally distributed >The population variances are equal ANOVA tests assume that variances can be pooled >The observations are independent 5 >We are interested in seeing of the advertising strategies employed in three cities made a difference >We assume that the three cities have been shown to be similar in the past >The sales results for 0 weeks in each of the three cities is displayed on the next slide 6

3 Data City City Quality City 3 Price Convenience Terminology >We have a response variable, the level of weekly sales >There are three factors or treatments, the advertising strategy used in the three cities 8 Means and Grand Mean City City Quality City 3 Price Convenience Mean Mean Mean Grand Mean

4 Discussion >There are differences between the means, but we are not sure if they are significant. >We could also observe that there is an amount of variation about the grand mean Some of this variation is explained by the treatments (advertising strategies) Some remains unexplained 0 Sum of Squares >In all forms of ANOVA, we analyze the SUMS OF SQUARES essentially, the numerator in the variance calculation Sum of Squares Between (SSB) SSB = k i= n ( x x) i i i =,, 3,, k group numbers > The difference between the each of the treatment (or factor) means and the grand mean is squared, multiplied by the number of responses for that treatment, and summed across treatments > If the treatment means equaled the grand mean, the SSB would be 0

5 Sum of Squares Within (SSW) SSW k nj = ( xij xi ) = i= j= ( n ) ( ) ( ) s + n s + n3 s3 > The unexplained variation, SSW, is sum of the residual variation around the treatment means > Since for each treatment, s = SS/(n-), we can also get the SSW by summing (n-) s for each treatment 3 Mean Squares SSB MSB = k SSW MSW = N k > The Mean Square for Treatments (i.e., between groups) is the SSB divided by the number of treatments minus > The Mean Square Within is the SSW divided by the sample size minus the number of treatments 4 The Test Statistic MSB F = MSW SSB MSB = k SSW MSW = N k > The ratio of the MSB divided by the MSW is distributed according to an F distribution, with: ν = df = (k - ) and ν = df = (N - k) 5

6 Means and Grand Mean City City Quality City 3 Price Convenience Mean Mean Mean Grand Mean City City City 3 Convenience Quality Price Mean Mean Mean Grand Mean Between Samples Grand Total (SSB) Within Samples s = 0775 s = 738. s 3 = Grand Total (SSW) MSB = F = MSW = p-value City City City 3 Convenience Quality Price Mean Mean Mean Grand Mean Between Samples Grand Total (SSB) Within Samples = 58. s = 0775 s = 738. s 3 = Grand Total (SSW) ( )( ) 7 MSB = F = MSW = p-value

7 Interpretation >Since P(F>3.3) = < α = 0.05, we reject H O : µ = µ = µ 3 >There is enough evidence to infer that the mean weekly sales differ between the cities. 9 ANOVA Table Standard ANOVA Table Source of Variation SS df Mean Square F-Statistic Between Samples SSB k - MSB = SSB/(k - ) F = MSB/MSW Within Samples SSW N - k MSW = SSW/(N - k) Total SST N - Source of Variation SS df Mean Square F-Statistic Between Samples 57, Within Samples 506, Total 564, Excel Output Anova: Single Factor SUMMARY Groups Count Sum Average Variance Convenience Quality Price ANOVA Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total

8 Required Conditions >Each treatment (sub-sample) must be normal and the variances equal >Our tests are crude: eyeball tests Looking at the histograms if not non-normal, assume normal text uses box and whisker plots Looking at the variances if not very different, assume the same Formulae: Single Factor ANOVA Source of Variation Between Groups SS df MS F SSB k Within Groups SSW N k Total SST N SSB MSB = k SSW MSW = N k MSB F = MSW 3 L3, Slides 5-7 Revisited >If we are simply looking at two samples, and want to see if there means are equal, we can perform the t-test we did in Lecture 4, or an F- test >This question was examining if there were differences Prof. Goodstat s morning and afternoon classes. 4

9 >Prof. Goodstat has two classes, one at 8:30 and one at :00. On the midterm, the morning class of 45 students had a mean of 70 and a standard deviation of, while the afternoon class of 40 had a mean of 75 and a standard deviation of 3. Is there evidence at α = 0.05 that the two classes are different? 5 t = ( x x ) ( µ µ ) ( 70 75) ( 0) s n s + n 5 = =.835 > Do Not Reject = s = p - If Pooled ( n ) s + ( n ) s ( 44) 44 + ( ) n + n ( x x ) ( µ µ ) ( 70 75) ( 0) t = = s p + n n 5 = =.8437 >.96.7 Do Not Reject = =

10 - Using ANOVA Morning Afternoon Overall Means Variances SS df MS F p-value Between Within When we did the example using a t-test, t = (t α/,df ) = F α,,df 8 Block Design 9 Terminology >Randomized Complete Block ANOVA (Text s terminology) >Two-way ANOVA without replication (Excel s terminology) >Other: Randomized Block Design Block design 30

11 Block Design >This is similar to the matched pair experiment, but with more than pairs We will have three or more treatments The matched pair can be viewed as a randomized block design with only two treatments 3 Plot Fertilizer A Fertilizer B Fertilizer C > Three fertilizers have been tested in 0 plots > The crop yields are shown at the left > We want to test for variation between fertilizers, but > We could have variation between the plots 3 >This is a two-way ANOVA without replication or block design since the researcher is controlling for differences that may exist between plots of land >Thus the first row (block) is represents the three different fertilizers in plot #, the second, plot #, etc. >Notice the similarity to Matched Pairs Design 33

12 >Columns (Fertilizers): H O : µ =µ =µ 3 H A : At least one is not equal α=0.0, Rejection region: F.0,,38 > 5. (Excel) >Rows (Plots): H O : All are equal H A : At least one is not equal α=0.0, Rejection region: F.0,9,38 >.4 (Excel) 34 >Note that if there are not significant differences between the blocks (rows) then the single factor test would be more appropriate Excel Output Anova: Two-Factor Without Replication SUMMARY Count Sum Average Variance Fert-A Fert-B Fert-C ANOVA Source of Variation SS df MS F P-value F crit Rows E Columns E Error Total

13 Explanation Source of Variation Rows Columns Error Total SS Variation attributable to the plots Variation attributable to the fertilizer Unexplained variation The total amount of variation to be explained 37 ANOVA Source of Variation SS df MS F Rows Columns Error Total >Since the F-Value (3.4) is greater than our critical F-Value (5.), we reject the null hypothesis that the fertilizers are the same >Likewise, the F-Value for the plots of land (33.) exceeds the critical value of.4 indicating it was appropriate to use this design >The same results can be inferred by the low p-values which are below our significance level, α=0.0 39

14 Discussion >We would not have been able to see the differences between the fertilizers, since the difference would have been lost in the variability between the plots. 40 Using One-way ANOVA Anova: Single Factor SUMMARY Groups Count Sum Average Variance Fertilizer A Fertilizer B Fertilizer C ANOVA Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total The Formulae: Block Design Source of Variation Between Groups Between Blocks Within Groups SS df MS F SSB k SSBL b SSB MSB = k SSBL MSBL = b SSW MSW = k b SSW (k )(b ) ( )( ) Total SST N MSB F = MSE MSBL F = MSE 4

15 Two-Factor ANOVA AKA Two-way ANOVA with replication 43 Two Factor ANOVA > extends Single Factor ANOVA >Suppose in the test market, we decide to investigate the impact of the type of media used: television and newspapers >Now we have two factors: The advertising message (before) The advertising medium (added here) 44 Hypotheses >For Message: H O : µ A = µ A = µ A3 H A : At least two means differ >For Media: H O : µ B = µ B H A : The two means differ 45

16 Data City- City- City-3 City-4 City-5 City-6 Convenience Quality Price TV NP TV NP TV NP Single Factor Test >We can perform the single factor test to see if there are differences between the cities. >Next slide, we see that there are differences between the cities 47 Anova: Single Factor Single Factor Test SUMMARY Groups Count Sum Average Variance Column Column Column Column Column Column ANOVA Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total

17 Two Factor Test >Knowing that there are differences between the cities we want to see if both factors are responsible for the differences 49 - Data Rearranged >We have to reorganize the data to reflect the two factors > The responses are coloured Yellow: Convenience and Television Blue: Quality and Television etc. Convenience Quality Price Television Newspaper Output - I Anova: Two-Factor With Replication SUMMARY Convenience Quality Price Total Television Count Sum Average Variance Newspaper Count Sum Average Variance Total Count Sum Average Variance

18 Output - ANOVA Table ANOVA Source of Variation SS df MS F P-value F crit Sample Columns Interaction Within Total > There is appears to be a difference between the messages > There is not enough evidence to suggest that the media or the interaction is significant 5 The Formulae: Two Factor Source of Variation SS df MS F Factor A SSA a Factor B SSB b Interaction SSAB (a )(b ) Error SSE N ab Total SST N SS A MS A = a SS B MS B = b SS AB MS AB = ( a )( b ) SSE MSE = N ab MS A F = MSE MSB F = MSE MS AB F = MSE 53 YOU LEARN STATISTICS BY DOING STATISTICS 54