August+3rd+slides - Quantitative Methods in Psychology...

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Unformatted text preview: Quantitative Methods in Psychology Psychology August 3rd, 2009: ANOVA week! Jeff Vietri corresponding to Chapter 13 in the corresponding text text This week Cover ANOVA Logic of ANOVA Only 2 chapters! How does it work? Why would we use it? Types of ANOVA One-way Repeated Multifactorial Today's agenda Go over quiz Introduce ANOVA Logic Terms Calculations? Don't forget the homework The problem What if we want to compare more than 2 groups? Z-test: sample vs. population Student's t-test: sample vs. population Independent-samples t-test: two samples Paired-samples t-test: one sample, two treatments, or two matched samples We haven't covered a test that can do that The problem The basis of the previous tests were mean differences, and so we could only use two values If we soldier ahead with multiple t-tests, we run into problems with experimentwise alpha error The risk of false alarms rises with each test, and the risk of making a single alpha error somewhere could get rather high The solution ANOVA, of course! Uses variance, rather than difference between the means The logic of ANOVA The variance among treatment means is due to effects of the treatments and chance/error/etc. The variability among scores within each treatment condition are due to chance/error/etc. If the variance between treatments is much bigger than the variance within treatments, the treatments did something ANOVA ANOVA Recall: Many Types of ANOVAs Between Ss = independent measures Within Ss = dependent measures Between subjects, within subjects, or mixed-models (both Between between and within Ss) between Single factor or multiple factors For now we will only consider single factor, For between subjects designs (chapter 13) between ANOVA: Hypotheses ANOVA: Null hypothesis H0: μ1 = μ2 = μ3 Populations behind all groups are the same Alternative hypothesis H1: there is at least one difference Not all populations represented by the groups are the Not same same ANOVA uses variance Recall: Variance is a measurement of difference Recall: among a set of numbers among The F-ratio The test statistic for the ANOVA is the Fratio How much variability is between my treatments / how much should I expect by chance For one-way ANOVA, this is variance btw / variance within F= = Treatment effect + Differences due to chance Difference due to chance Variability between treatments Variability within treatments F= Treatment effect + Differences due to chance Difference due to chance If there is no treatment effect (treatment effect = 0; null hypothesis is true): F= Treatment effect + Differences due to chance Differences due to chance F=1 If there is a treatment effect (treatment effect ≠ 0; null hypothesis is false): F= F>1 Treatment effect + Differences due to chance Difference due to chance ANOVA ANOVA The ANOVA uses an F-distribution (rather than a tdistribution or a z-distribution) F-distribution, like a t-distribution, is based on the F-distribution, degrees of freedom (df) (df) Bigger samples will give us better estimates of the Bigger population’s variances population’s Represents all the possible ratios (since the F is a Represents ratio of variances) ratio The tail end of the F-distribution represents the The critical region critical F-distribution is always positive Fcritical are different for each df Total Sum of Squares Total SSTotal = sum of squares for the entire set of scores (without regard to which sample scores come from) come SSTotal = ΣX SSTotal 2 ( ΣX ) − N G = Grand Total G = ΣX 2 G = ΣX − N 2 2 dftotal = N – 1 Within-groups Sum of Squares Sum SSwithin is the sum of squares for each group added together added SSwithin = ∑SSk = SS1 + SS2 + …+ SSk ∑SS within dfwithin = N – k – Remember “N” is total number of people in the Remember study and “k” in the number of samples study Between-groups Sum of Squares Between-groups SSBetween is the sum of squares of sample mean differences differences dfbetween = k – 1 (k: the number of samples) SS Between T12 T2 2 T3 2 G 2 − = + + n n2 n3 N 1 2 or , SSTotal 2 T G =Σ − n N T = total scores of a sample T1 = total scores of sample 1 n1 = number of people in sample 1 ANOVA ANOVA We now have SSTotal, SSWithin, SSBetween We now have dfTotal, dfWithin, dfBetween For variance, just as we are used to, we divide For SS by df to get variance SS So, for each type of SS, we divide the So, appropriate df to get a measure of variance df Except the result we call a mean square (or Except mean MS for short) rather than variance MS ANOVA Table ANOVA Source df SS MS F Between Groups (k – 1) SSB SSB/dfB MSB/MSW Within Groups (N – k) SSW SSW/dfW -- Total (N – 1) SSB + SSW -- -- F = MS between / MS within Official reporting Official If significant result: F(dfbetween, dfwithin) = F-value, p < α F-value, F(3, 12) = 6.25, p < .05 If non-significant result: F(3, 12) = 1.25, ns Example Example Control 3 0 2 0 0 T=5 SS = 8 Drug 1 Drug 2 4 6 3 3 1 4 1 3 1 4 T=10 T=20 T=25 SS= 8 SS= 6 n = 20 G =60 ΣX2 = 262 Drug 3 7 6 5 4 3 SS=10 Exercise Exercise Soda A 0 4 0 1 0 T=5 SS = 12 Soda B 6 8 5 4 2 T=25 SS= 20 n = 15 G = 60 ΣX2 = 356 Soda C 9 5 6 6 4 T=30 SS= 14 SS= ANOVA ANOVA If the F-test is significant this tells us that If there are differences between our groups there However, we may want to know exactly However, which groups are different which We can only assume that the sample with the We largest mean is significantly different than the group with the smallest mean group Post Hoc Tests Post Post hoc means “after this” These tests are conducted after the initial ANOVA These to find out which groups are significantly different from one another from These tests are only done when: These only The null hypothesis is rejected (a significant F ratio) AND you have three or more groups (if you just had two AND groups, you wouldn’t need post hoc tests because you would know the group with the highest mean is different from the group with the lowest) from Tukey’s HSD Tukey’s MS within HSD = q n1 MSwithin is the same value from conducting the ANOVA ANOVA q is calculated from the Studentized Range Statistic is table (Page 696) table Use number of groups (total samples; k) and dfwithin The “n1” in the formula refers to the number of people in in each sample (it assumes there are equal numbers) each Tukey’s HSD Tukey’s The value of HSD indicates the amount two sample The means must differ by to be considered significantly different different Once you have the HSD value, calculate the Once differences in sample means for each comparison you are interested in (µ1- µ2; µ1- µ3 ; µ2- µ3) you Each difference is compared to the HSD value, if Each that difference is greater than the HSD value then those groups are significantly different those Effect Size Effect A significant effect is not always a big effect We use measures of effect size to determine We the relative impact of the IV on the DV the With t-tests we used Cohen’s d and r2 With Cohen’s With ANOVA we use η2 With Called “eta-squared” Effect Size Effect SSbetween η= SSbetween + SS within 2 or SSbetween η= SS total 2 Effect Size Effect SSbetween η= SSbetween + SS within 2 Gives the percent variance explained (just like Gives r2) 0.01 < η2 < 0.09 small effect 0.01 0.09 < η2 < 0.25 medium effect 0.09 η2 > 0.25 large effect ...
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This note was uploaded on 11/17/2011 for the course PSYCHOLOGY 830:200 taught by Professor Staff during the Fall '11 term at Rutgers.

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