Compare Multiple Response Variables

Transcription

1 Compare Multiple Respose Variables STATGRAPHICS Mobile Rev. 4/7/006 This procedure compares the data cotaied i three or more Respose colums. It performs a oe-way aalysis of variace to determie whether or ot there are sigificat differeces betwee the colum meas, ad also coducts a test to compare the variaces withi each colum. Tukey s HSD multiple compariso method is used to idetify which colum meas are sigificatly differet tha which other colum meas. The data for this aalysis cosist of samples from m populatios. Let i = i-th measuremet i sample = size of sample Access Highlight: 3 or more Respose colums. Select: Compare from the mai meu. Output Page 1: A scatterplot of the data i each sample, together with summary statistics. Output Page : Bo-ad-whisker plots for the data i each sample, together with the results of statistical tests that compare the samples. Output Page 3: A plot of the sample meas with Tukey s HSD itervals (or other selected ucertaity itervals). Output Page 4: A plot of the variability withi each group. Sample Data The file widgets.sgm cotais data o the stregth of widgets made from m = 4 differet types of material. = 1 widgets were samples from each of the four materials. A portio of the data is show below. Mat A Mat B Mat C Mat D by StatPoit, Ic. Compare Multiple Respose Variables - 1

2 STATGRAPHICS Mobile Rev. 4/7/006 Scatterplot The Scatterplot displays the data i each colum. The poits are ittered (radomly off i the horizotal directio) to help prevet overplottig. Also displayed is the umber of observatios i up to 8 colums. 006 by StatPoit, Ic. Compare Multiple Respose Variables -

3 STATGRAPHICS Mobile Rev. 4/7/006 Boplots The Boplots page displays a bo-ad-whisker plot for the data i each colum. The plot is costructed i the followig maer: A bo is draw etedig from the lower quartile of each sample to the upper quartile. This is the iterval covered by the middle 50% of the data values whe sorted from smallest to largest. A horizotal lie is draw at the media (the middle value). A plus sig is placed at the locatio of the sample mea. Whiskers are draw from the edges of the bo to the largest ad smallest data values. Below the plot are the results of four statistical tests: (1) A oe-way aalysis of variace to compare the meas of the populatios from which the samples were take. The output displays: 006 by StatPoit, Ic. Compare Multiple Respose Variables - 3

4 STATGRAPHICS Mobile Rev. 4/7/006 MSB: mea square betwee colums, which estimates the variace betwee the colums. It is calculated from the deviatios of the group mea aroud the grad mea: MSB = m = 1 ( ) m 1 (1) where the group meas are give by i i i= = 1 () ad the grad mea is m i = i= = m i 1 1 (3) = 1 MSE: mea square error, which estimates the variace withi the colums. It is calculated from the deviatios of the observatios aroud their respective colum meas: MSE = m ( i ) = 1 i= 1 = 1 m (4) F-ratio: the ratio of the mea squares MSB F = (5) MSE P-value: a P-value calculated by referrig the calculated F-ratio to Sedecor s F distributio. A small value of P idicates that there are sigificat differeces amogst the group meas. I the sample data, the differece is etremely sigificat. () Levee s test, which is used to determie whether or ot the group variaces are sigificatly differet. Levee s test performs a oe-way aalysis of variace o the variables z i = (6) i 006 by StatPoit, Ic. Compare Multiple Respose Variables - 4

5 STATGRAPHICS Mobile Rev. 4/7/006 The tabulated statistic is the F statistic from the resultig ANOVA table. A P-value above 0.05, such as that displayed for the sample data, meas that there is ot a statistically sigificat differece amogst the group variaces at the 5% sigificace level. (3) The Kruskal-Wallis test, which compares the group medias by cobiig all of the observatios ito a sigle sample, rakig them, ad the calculatig the average rak for each group. A small P-value, such as that illustrated for the sample data, idicates that the populatios medias are sigificatly differet from oe aother. All of the tests here assume that the data are samples from ormal distributios. To select a differet distributio: 1. Access the Properties dialog bo for the leftmost Respose variable by double-clickig o its colum header.. O the Dist. tab, select the assumed distributio. If you select Logormal, the logarithms of the data will be assumed to follow a ormal distributio. If you select Power ormal, the data will be assumed to follow a ormal distributio after raisig them to the idicated power. 006 by StatPoit, Ic. Compare Multiple Respose Variables - 5

6 STATGRAPHICS Mobile Rev. 4/7/006 If a o-ormal distributio is selected, the tests will be applied to the data i the trasformed metric. Meas Plot The Meas Plot page displays the mea of each row as a poit symbol, together with Tukey s HSD (Hoestly Sigificat Differece) itervals. The poit symbols are located at the sample meas. The bars eted over ucertaity itervals, the type of which is determied by the settigs o the Comp. tab of the leftmost Respose colum. Several differet types of itervals may be costructed: Stadard errors (pooled s) - displays the stadard errors usig the pooled withi-sample stadard deviatio: MSE ± (7) 006 by StatPoit, Ic. Compare Multiple Respose Variables - 6

7 STATGRAPHICS Mobile Rev. 4/7/006 Stadard errors (idividual s) - displays the stadard errors usig the stadard deviatio of each sample separately: s ± (8) Cofidece itervals (pooled s) - displays cofidece itervals for the group meas usig the pooled withi-group stadard deviatio: MSE tα (9) ± /, m where equals the total umber of observatios. Cofidece itervals (idividual s) - displays cofidece itervals for the sample meas usig the stadard deviatio of each group separately: s ± t α /, 1 (10) LSD itervals - desiged to compare the meas with the stated cofidece level. The itervals are give by ± t α /, m s p (11) where t represets the value of Studet s t distributio with + q degrees of freedom leavig a area of α/ i the upper tail of the curve. If two sample sizes are equal, you ca determie whether or ot the populatio meas are sigificatly differet from each other by determiig whether or ot the itervals overlap. Tukey HSD Itervals - desiged for comparig multiple pairs of meas.. The itervals are give by ± T α /, q, q s p (1) which uses Tukey s T istead of Studet s t. Tukey called his procedure the Hoestly Sigificat Differece procedure sice it allows all pairs of meas to be compared while cotrollig the eperimet-wide error rate at α. If all of the meas are equal, the probability of icorrectly declarig ay of the pairs to be sigificatly differet i the etire eperimet equals α. Tukey s procedure is more coservative tha Fisher s LSD procedure, sice it makes it harder to declare ay particular pair of meas to be sigificatly differet. 006 by StatPoit, Ic. Compare Multiple Respose Variables - 7

8 STATGRAPHICS Mobile Rev. 4/7/006 Residuals The Residuals plots the deviatios of each observatio from its respective group mea. The plot may be used to help idetify outliers, ad to visualize ay differeces betwee the withi-group variace. The sample stadard deviatio for up to 8 groups is displayed below the plot. 006 by StatPoit, Ic. Compare Multiple Respose Variables - 8