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1 STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence rather than goodness of ft tests, where we are testng to see whether or not two varables are related or dependent, n partcular, whether or not the mean of two samples equals. For example, two groups ( x 1 = ( u 1, u 2,..., um), x 2 = ( v 1, v 2,..., vn) ) each receve a dfferent teachng method, m and ther average fnal exam scores ( y1 = u / m, y2 = v / n ) are compared. The logc s: f the two = 1 = 1 groups do not dffer n average exam scores, we conclude that there s no relatonshp between teachng method and fnal exam scores. Two sample Z-test The general form of the formula s the same as for the one sample z test: z equals the dfference of the test statstc and the null hypothess value (observed dfference expected dfference) dvded by the standard devaton of the statstc. In the case of two samples, we smply subtract the means (and dvde by the standard error of the dfference between means) to obtan the test statstc. Below s the full z formula: n Z ( y y ) ( µ µ ) y y σ σ = = and y1 y2 y1 y2 2 2 σ σ σ = σ + σ = + y1 y2 y1 y2 m n Note how the z formula smplfes when the dfference of populaton means equals 0 ( µ 1 µ 2 = 0 as t does n most cases). Agan, f the populaton standard devatons ( σ1, σ 2 ) are unknown, but sample sze s suffcently large, we can stll use ths formula by substtutng the sample standard devatons for the populaton standard devatons. Two sample (ndependent) t-test The formula for the two sample t test closely matches the formula for the two sample z test: t ( y y ) ( µ µ ) y y S S df = =, wth degrees of freedom ( df ) ( m 1) ( n 1) m n 2 y1 y2 y1 y2 = + = +. Here, the standard error of the dfference between means can be calculated drectly from the data. (We assume that the samples are from two populatons wth same varance. Ths formula provdes the unbased estamte we are lookng for!) m n 2 2 ( u y1 ) + ( 2 ) v y = 1 = 1 S = y1 y2 1 1 ( + ) ( m 1) + ( n 1) m n.
2 Permutaton t-test Please read the paper by Dudot et al or the notes taken from lecture. The paper wll be dstrbuted n class. ANOVA (analyss of varance), F-test The nature and purpose of analyss of varance: Analyss of varance s a generalzaton of the t-test o The t-test s used to test the null hypothess: µ 1 =µ 2 o The t-test s approprate for only two groups o Analyss of varance, and the assocated F-test, can test for dfferences between any number of means for categorzed varables. o Formally, the null hypothess s µ 1 = µ 2 = µ 3 = µ 3... = µ n The underlyng structural model of Analyss of varance: Each observaton n the combned dstrbuton represents a lnear combnaton of components: X j = µ grand + a j + e j Where: a j = effects of group and e j = effects of random error If the treatment or "group" effect s 0, o then observatons wll vary around the mean dependng on e j (errors around the mean) o these errors are assumed to be random, normally dstrbuted, and sum to 0. Ths model--whch accounts for ndvdual observatons n the dstrbuton--can be expressed n terms of the total varaton of ndvdual observatons from the grand mean. Parttonng the total varaton (also known as sum of squares, SS): More succnctly: TSS=WSS+BSS Calculaton of sums of squares: An ntutve approach to what s gong on: If the group means were really dfferent, then the BSS (explaned SS) would be large relatve to the WSS (unexplaned SS). If the group means were not much dfferent, the WSS would be large relatve to the BSS. BSS and WSS as tests of populaton varance: Each of these provde the bass for two ndependent estmates of the populaton varance o One estmates the populaton varance based on the varance wthn each of the samples o The other estmates the populaton varance based on varance between the sample means The F test s smply a rato between these two estmates of the populaton varance: estmate of varance based on between mean varaton F = estmate of varance based on wthn group varaton
3 If ths rato s small, then the two estmates agree closely, and we conclude that the groups represent random samples from equvalent populatons:.e., same means and varances How do we get these estmates of the populaton varances? They must be dvded by the approprate degrees of freedom o BSS / k-1 = between groups mean square (read "mean square" as the mean of the sum of squares) o WSS / N-k = wthn groups mean square Quantle-Quantle Plot The quantle-quantle (q-q) plot s a graphcal technque for determnng f two data sets come from populatons wth a common dstrbuton. A q-q plot s a plot of the quantles of the frst data set aganst the quantles of the second data set. By a quantle, we mean the fracton (or percent) of ponts below the gven value. That s, the 0.3 (or 30%) quantle s the pont at whch 30% percent of the data fall below and 70% fall above that value. A 45-degree reference lne s also plotted. If the two sets come from a populaton wth the same dstrbuton, the ponts should fall approxmately along ths reference lne. The greater the departure from ths reference lne, the greater the evdence for the concluson that the two data sets have come from populatons wth dfferent dstrbutons. Above q-q plot shows that 1. These 2 batches do not appear to have come from populatons wth a common dstrbuton. 2. The batch 1 values are sgnfcantly hgher than the correspondng batch 2 values. The dfferences are ncreasng from values 525 to 625. Then the values for the 2 batches get closer agan. The q-q plot s formed by: Vertcal axs: Estmated quantles from data set 1 Horzontal axs: Estmated quantles from data set 2 Both axes are n unts of ther respectve data sets. That s, the actual quantle level s not plotted. For a gven pont on the q-q plot, we know that the quantle level s the same for both ponts, but not what that quantle level actually s. If the data sets have the same sze, the q-q plot s essentally a plot of sorted data set 1 aganst sorted data set 2. If the data sets are not of equal sze, the quantles are usually pcked to correspond to the sorted values from the smaller data set and then the quantles for the larger data set are nterpolated.
4 The q-q plot can answer the followng questons: Do two data sets come from populatons wth a common dstrbuton? Do two data sets have common locaton and scale? Do two data sets have smlar dstrbutonal shapes? Do two data sets have smlar tal behavor? When there are two data samples, t s often desrable to know f the assumpton of a common dstrbuton s justfed. If so, then locaton and scale estmators can pool both data sets to obtan estmates of the common locaton and scale. If two samples do dffer, t s also useful to gan some understandng of the dfferences. The q-q plot can provde more nsght nto the nature of the dfference than analytcal methods such as the ch-square and Kolmogorov-Smrnov 2-sample tests. The normal Q-Q plot graphcally compares the dstrbuton of a gven varable to the normal dstrbuton (represented by a straght lne). The advantages of the q-q plot are: 1. The sample szes do not need to be equal. 2. Many dstrbutonal aspects can be smultaneously tested. For example, shfts n locaton, shfts n scale, changes n symmetry, and the presence of outlers can all be detected from ths plot. The q-q plot s smlar to a probablty plot. For a probablty plot, the quantles for one of the data samples are replaced wth the quantles of a theoretcal dstrbuton. R commands related to Q-Q plots: >qqnorm(x) # a quantle-quantle plot of the data sample x aganst standard normal #dstrbuton. >qqplot(x,y) # a quantle-quantle plot of two data samples x and y. >quantle(x, probs=seq(0,1,0.25)) # produces sample quantles correspondng to the gven probabltes. The # smallest observaton corresponds to a probablty of 0 and the largest to a #probablty of 1. In the above command, you can try to change 0.25 to #other small numbers n (0,1) to get dfferent quantles. Correlaton and regresson The followng table records the heght of 60 fathers and ther sons: Let X =heght of father, Y =heght of son. How are X and Y related?
5 The fgure below shows the scatter dagram for the heghts of 60 fathers and sons. Each pont n the fgure corresponds to a par ( X, Y ). The scatter dagram n fgure 1 s a cloud shaped somethng lke a football. 85 Son's heght (nches Father's heght (nches) The swarm of ponts n fgure 1 slopes upward to the rght, the y-coordnates of the ponts tendng to ncrease wth ther x-coordnates. So there s a postve assocaton between the heghts of fathers and sons. As a rule, the taller fathers have taller sons. Ths confrms the obvous. The scatter plot s most nformatve for two varables. In studes of the relatonshp between two varables, t s usual to label one as ndependent and the other as dependent. Ordnarly, the ndependent varable s thought to nfluence the dependent varable, rather than the other way around. In fgure 1, father s heght s taken as the ndependent varable and plotted along the x-axs: father s heght nfluences son s heght. However, there s nothng to stop an nvestgator from usng son s heght as ndependent varable. Pearson correlaton coeffcent to measure LINEAR relatonshp: 1 1 ( x x)( y y) ( x y ) x y rxy = n = n ( x x) / n ( y y) / n ( x x) / n ( y y) / n, where x, y are the means. Propertes of correlaton coeffcent: r = r. 1. xy yx 2. 1 r xy If v = a + bx, w = c + d y, 4. If y = a + bx, r xy r vw 1, b > 0 = 0, b = 0 1, b < 0 b d = r b d xy
6 Interpretatons on correlaton coeffcent r 1. the closer r to 1, the easer to predct y from x or vce versa. 2. r > 0, postve assocaton. The bgger x, the bgger y on average. 3. r < 0, negatve assocaton. The bgger x, the smaller y on average. 4. correlaton causaton. Independent varable x does not have to cause the dependent varable y. 5. r does not measure nonlnear relatonshp. Predcton based on assocaton Ideal case for lnear predcton from x to y (or y to x): the scatter plot shows a football shape. Summary on the data for heghts of 60 fathers and sons: x = nches, SD(x)=3.78 nches; y = 69.73, SD(y)=3.75 nches r = Q: Gven a father s heght x = x0, how would you guess the son s heght y 0 usng the nformaton n the data set? Suppose you have access to every data pont n the set, what s a good guess? Take a vertcal strp around x = x0, get the average heght of ponts n that strp, and ths s the guessed or predcted heght of the son whose father has the heght of x = x0. If we draw many such vertcal strps at dfferent values of heght of fathers, the average of strps happen to fall on a lne (more or less): Ths s called the REGRESSION LINE. Propertes of the regresson lne: 1. regresson lne passes through ( x, y ) SD( y) 2. Slope of regresson lne = r. SD ( x ) 3. When r 0, the regresson lne s almost horzontal passng through ( x, y ), so guess s roughly y ; knowng x 0 s not so helpful. Q: Should we use the same lne to predct x from y? SD( x) Horzontal strps lead to another regresson lne wth slope r. SD ( y )
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