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Chapter 13. Assumptions Underlying Parametric Statistical Techniques. Parametric Statistics. We have been studying parametric statistics. They include estimations of mu and sigma, correlation, t tests and F tests. Five Assumptions. two research assumptions;

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## Chapter 13

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**Chapter 13**Assumptions Underlying Parametric Statistical Techniques**Parametric Statistics**• We have been studying parametric statistics. • They include estimations of mu and sigma, correlation, t tests and F tests.**Five Assumptions**• two research assumptions; • two assumptions about the type of the distributions in the samples, • and one assumption about the kind of numbering system that we are using. To validly use parametric statistics, we make**Research Assumptions**• Subjects have to be randomly selected from the population. • Experimental error is randomly distributed across samples in the design. (We will not discuss these any further).**Distribution Assumptions**• The distribution of sample means fit a normal curve. • Homogeneity of variance (using FMAX).**Assumptions about Numbering Schemes**• The measures we take are on an interval scale. (Other numbering scales, such as ordinal and nominal, do not allow the estimation of population parameters such as mu and sigma and the tests used to analyze such data are therefore call “nonparametric”).**Violating the Assumptions**If any of these assumptions are violated, we cannot use parametric statistics. We must use less-powerful, non-parametric statistics.**Sample Means**• An assumption we need to make is that the distribution of sample means is normally distributed. • This is not as extreme an assumption as it might seem. • We will follow the example in the book to demonstrate (only smaller).**An Artificial Population**Subject Score • Seven subjects. • Each subject has a different score. • We sample five subjects. A 1 B 2 C 3 D 4 E 5 F 6 G 7**The Distribution is Rectangular**FREQUENCY 3 2 1 0 1 2 3 4 5 6 7 SCORE**All Possible Samples**Sample Scores Mean Sample Scores Mean ABCDE 12345 3.0 ABCDF 12346 3.2 ABCDG 12347 3.4 ABCEF 12356 3.4 ABCEG 12357 3.6 ABCFG 12367 3.8 ABDEF 12456 3.6 ABDEG 12457 3.8 ABDFG 12467 4.0 ABEFG 12567 4.2 ACDEF 13456 3.8 ACDEG 12457 4.0 ACDFG 13467 4.2 ACEFG 13567 4.4 ADEFG 14567 4.6 BCDEF 23456 4.0 BCDEG 23457 4.2 BCDFG 23467 4.4 BCEFG 23567 4.6 BDEFG 24567 4.8 CDEFG 34567 5.0**Normal Curve for Sample Means Conclusion**Even if we have a small population (7), … with a rectangular distribution, … and a small sample size (5), … which yields a small number of possible samples (21), … the sample means tend to fall in an (approximately) normal distribution. This assumption that the distribution of sample means will basically fit a normal curve is seldom violated. This assumption is robust.**But it can happen -Violating the Normal Curve Assumption**Distributions of sample means can vary from normal in several ways. Normal curves • are symmetric • are bell-shaped • have a single peak Non-normal curves • have skew • have kurtosis- platykutic or leptokurtic • are polymodal**The left side**is the same shape as the right side. Symmetry F r e q u e n c y score**Skewed Right**Skewed Left Skewed NORMAL**Area under the curve occurs in a prescribed manner,**as listed in the Z table. Bell-shaped F r e q u e n c y 1 SD is 34%; 2 SD is 48%; etc. score**Platykurtic**Leptokurtic Kurtosis NORMAL**One mode**F r e q u e n c y score There is only one mode and it equals the median and the mean.**Trimodal**Bimodal Polymodality NORMAL**Violation of normally distributed sample means**If the distribution of sample means is • … skewed, • … or has kurtosis, • … or more than one mode, • … then we cannot use parametric statistics. • BUT THIS IS RARE.**For F Ratios and t Tests**• We assume that the distribution of scores around each sample mean is similar. • The distributions within each group all estimate the same thing, that is, sigma2. • The mean squares within each group should be the approximately the same in each group, differing only because of random sampling fluctuation. • For F ratios and t tests, this is called homogeneity of variance.**For Correlation**• For correlation, the scores must vary roughly the same amount around the entire length of the regression line. • This is called homoscedasticity.**3**2 1 -3 -2 -1 0 1 2 3 0 -1 -2 -3 Homoscedasticity**3**2 1 -3 -2 -1 0 1 2 3 0 -1 -2 -3 Non-Homoscedasticity**Homogeneity of Variance**In mathematical terms, homogeneity of variance means that the mean squares for each group are the same. We use the FMAX test to check if the group with the smallest mean squares is “too different” from the group with the largest mean squares.**FMAX**• If FMAX is significant, then the Mean Squares deviate from each other too much. • The assumption of homogeneity of variance is violated. • We cannot use parametric statistics!**Why???**• Because all parametric statistical procedures rely on our ability to estimate sigma2 with MSW. • If the estimates of MSW among the grous differ among groups so that Fmax is significant, the odds are someone (most likely the senior experimenter) messed up and created a measure with too small a range of scores.**When that happens all the scores pile up at one end of the**scale. • When everyone scores at the top or bottom a scale, individual differences and measurement problems seem to disappear. • We call this a ceiling effect (if the scores are all at the top of the scale) and a floor effect if the scores are all at the bottom**Because ID and MP in one or more groups have been pushed up**against the top or bottom of the scale there is practically no within group variation. • So, while adding df, the group contributes little or nothing to sum of squares within group (SSW). • So, when you include one or more groups with practically no variation within group in your totals sums of squares and mean square, you wind up with an underestimate of sigma2. • This makes it possible to get significant results not because you have pushed the means apart with an IV, but because MSW is an underestimat**This makes it possible to get significant results not**because you have pushed the means apart with an IV, but because MSW is an underestimate of sigma2and therefore the denominator of the F or t test will be too small. • So you can get significant results more often than you should when the null is true.**4.00**4.00 4.00 4.00 4.00 4.00 4.00 4.00 -3.00 -2.00 -1.00 1.00 1.00 2.00 2.00 0.00 9.00 4.00 1.00 1.00 1.00 4.00 4.00 0.00 8.75 8.75 8.75 8.75 8.75 8.75 8.75 8.75 .25 -.75 .25 .25 .25 .25 -.75 .25 .06 .56 .06 .06 .06 .06 .56 .06 Calculate the means. Calculate the deviations. Square the deviations. Sum the deviations. Divide by df NG-1. Uncrowded vs crowded groups – How crowded do you feel? 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1 2 3 5 5 6 6 4 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 9 8 9 9 9 9 8 9**FMAX**In FMAX, the “MAX” part refers to the largest ratio that can be obtained by comparing the estimated variances from 2 experimental groups. The significance of FMAX is checked in an FMAX table.**The number of groups**in the experiment. nG(larger) - 1 Interpolate to larger df. K = number of variances 2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX alpha = .01.**The critical values.**k = number of variances 2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX**k = number of variances**2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX FMAX = 16.33 > 8.89 FMAX exceeds the critical value. We cannot use parametric statistics.**Examples**Number Subjects Critical value Design of Means in larger NG of FMAX 2X4 8 21 5.3 2X2 ? 16 ? 3X3 ? 11 ? 2X3 ? 9 ? 4 9 6**K = number of variances**2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX**Number Subjects Critical value**Design of Means in larger NG of FMAX 2X4 8 21 5.3 2X2 4 16 5.5 3X3 9 11 ? 2X3 6 9 ?**K = number of variances**2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX**Number Subjects Critical value**Design of Means in larger NG of FMAX 2X4 8 21 5.3 2X2 4 16 5.5 3X3 9 11 12.4 2X3 6 9 ?**K = number of variances**2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX**Number Subjects Critical value**Design of Means in larger NG of FMAX 2X4 8 21 5.3 2X2 4 16 5.5 3X3 9 11 9.5 2X3 6 9 14.5**Example – other way**Number of Means 8 ? ? ? MSG max 18.2 26.3 34.2 18.0 MSG min 1.1 2.0 4.6 0.5 FMAX 16.5 ? ? ? Subjects in larger NG 10 12 21 7 dfFMAX 9 ? ? ? p.01 .01 ? ? ? Design 2X4 2X3 2X2 3X3 11 6 13.2 20 4 7.4 6 9 36.0**FMAX(6,11) = 13.2**k = number of variances 2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX p.01**Number**of Means 8 ? ? ? MSG max 18.2 26.3 34.2 18.0 MSG min 1.1 2.0 4.6 0.5 FMAX 16.5 13.2 7.4 36.0 Subjects in larger NG 10 12 21 7 dfFMAX 9 11 20 6 p.01 .01 .01 ? ? Design 2X4 2X3 2X2 3X3 6 4 9**FMAX(4,20) = 7.4**k = number of variances 2 3 4 5 6 7 8 9 10 4 23.2 37 49 59 69 79 89 97 106 5 14.9 22 28 33 38 42 46 50 54 6 11.1 15.5 19.1 22 25 27 30 32 34 7 8.89 12.1 14.5 16.5 18.4 20 22 23 24 8 7.50 9.9 11.7 13.2 14.5 15.8 16.9 17.9 18.9 9 6.54 8.5 9.9 11.1 12.1 13.1 13.9 14.7 15.3 10 5.85 7.4 8.6 9.6 10.4 11.1 11.8 12.4 12.9 12 4.91 6.1 6.9 7.6 8.2 8.7 9.1 9.5 9.9 15 4.07 4.9 5.5 6.0 6.4 6.7 7.1 7.3 7.5 20 3.32 3.8 4.3 4.6 4.9 5.1 5.3 5.5 5.6 30 2.63 3.0 3.3 3.4 3.6 3.7 3.8 3.9 4.0 60 1.96 2.2 2.3 2.4 2.4 2.5 2.5 2.6 2.6 dfFMAX p.01

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