Lesson 15 ANOVA (analysis of variance)


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1 Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi F (ratio of betwee to withi Example Problem Lesso 5 ANOVA (aalysis of variace Note: The formulas detailed here vary a great deal from the text. I suggest usig the otatio I have outlied here sice it will coicide more with what we have already doe, but you might loo at the text versio as well. Use whatever method you fid easiest to uderstad. Variability Please read about this topic o the web page by locatig the ANOVA demostratio or you ca clic here: Betwee group variability ad withi group variability are both compoets of the total variability i the combied distributios. What we are doig whe we compute betwee ad withi variability is to partitio the total variability ito the betwee ad withi compoets. So: Betwee variability withi variability total variability Hypothesis Testig Agai, with ANOVA we are testig hypotheses that ivolve comparisos of two or more populatios. The overall test, however, will idicate a differece betwee ay of the groups. Thus, the test will ot specify which two, or if some or all of the groups differ. Istead, we will coduct a separate test to determie which specific meas differ. Because of this fact the research hypothesis will state simply that at least two of the meas differ. The ull will still state that there are o sigificat differeces betwee ay of the groups (isert as may mu s as you have groups. H 0 : µµµ3 Critical values are foud usig the Ftable i your boo. The table is discussed i the example below. Computatio How do we measure variability i a distributio? That is, how do we measure how differet scores are i the distributio from oe aother? You should ow that we use variace as a measure of variability. With ANOVA or aalysis of variace, we compute
2 a ratio of variaces: betwee to withi variace. Recall that variace is the average square deviatio of scores about the mea. We will compute the same value here, but as the defiitio suggests, it will be called the mea square for the computatios. So, we are computig variace. Recall that whe we compute variace we first fid the sum of the square deviatios, ad the divide by the sample size (  or degrees of freedom for a sample. s ( X X Sums _ of _ Squares deg rees _ of _ freedom Whe we compute the Mea Square (variace i order to form the Fratio, we will do the exact same thig: compute the sums of squares ad divide by degrees of freedom. Do t let the formulas itimidate you. Keep i mid that all we are doig is fidig the variace for our betwee factor ad dividig that by the variace for the withi factor. These two variaces will be computig by fidig each sums of squares ad dividig those sums of squares by their respective degrees of freedom. Sums of Squares We will use the same basic formula for sums of squares that we used with variace. While we will oly use the betwee variace ad withi variace to compute the F ratio, we will still compute the sums of squares total (all values for completeess. Total Sums of Squares X X ( N Note that it is the same formula we have bee usig. The subscript (tot stads for the total. It idicates that you perform the operatio for ALL values i your distributio (all subjects i all groups. Withi Sums of Squares ( X ( X ( X X X X... Notice that each segmet is the same formula for sums of squares we used i the formula for variace ad for the total sums of squares above. What is differet here is that you cosider each group separately. So, the first segmet with the subscript meas you compute the sum of squares for the first group. Group two is labeled with a, but otice that after that we have group istead of a umber. This otatio idicates that you cotiue to fid the sums of squares as you did for the first two groups for however
3 may groups you have i the problem. So, could be the third group, or if you have four groups the you would do the same sums of squares computatio for the third ad fourth group. Betwee Sums of Squares ( X ( X ( X ( X... N We have the same otatio here. Agai, you perform the same operatio for each separate group i your problem. However, with this formula oce we compute the value for each group we must subtract a operatio at the fial step. This operatio is half the sums of squares we computed for the sums of squares total. Degrees of Freedom Agai, we will first compute the sums of squares for each source of variace, divide the values by degrees of freedom i order to get the two mea square values we eed to form the Fratio. Degrees of freedom, however, is differet for each source of variability. Total Degrees of freedom N this N value is the total umber of values i all groups Withi Degrees of freedom N K K is the umber of categories or groups, N is still the total N withi Betwee Degrees of freedom Betwee K We will also use degrees of freedom to locate the critical value o the Ftable (see page A9 for alpha.05 ad A30 for alpha.0. The umerator of the Fratio is the betwee factor, so we will use the degrees of freedom betwee alog the top of your table. The deomiator of the Fratio is the withi subjects factor, so will use degrees of freedom withi alog the left margi of the table. Mea Square Now we divide each sums of squares by the respective mea square. Do t let the formula s itimidate you. All we are doig is matchig up degrees of freedom with the Sums of squares to get the mea square (variace Withi Mea Square Betwee Betewee Betwee Betwee Mea Square Withi Withi
4 Fratio The fial step is to divide our betwee by withi variace to see if the effect (betwee is large compared to the error (withi. Betwee F Example Withi A therapist wats to examie the effectiveess of 3 therapy techiques o phobias. Subjects are radomly assiged to oe of three treatmet groups. Below are the rated fear of spiders after therapy. Test for a differece at α.05 Therapy A Therapy B Therapy C Σ x 8 Σ x 0 Σ x 3 5 Σ 74 Σ 6 x STEP : State the ull ad alterative hypotheses. H at least oe mea differs H 0 : µ µ µ3 x Σ x 3 7 STEP : Set up the criteria for maig a decisio. That is, fid the critical value. You might do this step after Step 3 sice that is where you compute the critical value. Betwee K 3 withi F critical 3.88 N K 53 STEP 3: Compute the appropriate teststatistic. Although i this example I have give the summary values, for some problems you might have to compute the sum of x, ad sum of squared x s yourself. ( X X ( N
5 ( X ( X ( X X X X... ( 8 ( 0 ( ( ( 6 0 ( ( X ( X ( X ( X ( 8 ( 0 ( 5 ( N Note that aytime you compute two of the Sums of Squares you ca derive the third oe without computatio because Betwee Withi Total tot N Betwee K withi N K tot withi 5 3 Betwee Betwee Withi Betewee F Betwee Withi Betewee F 6.43 Betwee Oce we have computed all the values, very ofte we place them i a source table (below. Puttig the values i a table lie this oe may mae it easier to thi about the statistic. Notice that oce we get the Sums of Squares o the table, we will divide those values by the i the ext colum. Oce we get the two mea squares we divide those to get F. Withi
6 Source F Betwee (group _ Withi (error _7..43_ Total _ STEP 4: Evaluate the ull hypothesis (based o your aswers to the above steps. Reject the ull STEP 5: Based o your evaluatio of the ull hypothesis, what is your coclusio? There is at least oe group that is differet from at least oe other group.
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