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26/06/2008 -1- Praktische Statistik für Umwelt- und Geowissenschaftler Uni/bivariate Probleme Parametrische VerfahrenNicht parametrische Verfahren Unabhängigkeit.

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Präsentation zum Thema: "26/06/2008 -1- Praktische Statistik für Umwelt- und Geowissenschaftler Uni/bivariate Probleme Parametrische VerfahrenNicht parametrische Verfahren Unabhängigkeit."—  Präsentation transkript:

1 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Uni/bivariate Probleme Parametrische VerfahrenNicht parametrische Verfahren Unabhängigkeit NormalverteilungAusreißer (Phasen-Iterationstest) (KS-Test / Chi-Quadrat Test)(Dixon / Chebyshev's Theorem) Verteilungstest Vergleich von Mittelwerten mit dem Parameter der GG Zusammenhangsanalyse Voraussetzungen erfüllt ? Vergleich von 2 unabhängigen Stichproben Vergleich von k unabhängigen Stichproben KS-Test / Chi-quadrat Test H-Test Rangkorrelation nach Spearmann Einstichproben T-test Chi-quadrat Test Zweistichproben T-test F-Test / Levene Test Varianzanalyse (ANOVA) Pearsons Korrelationsanalyse/ Regressionsanalyse Vergleich von 2 verbundenen Stichproben Wilcoxon-Test U-Test T-Test für verbundene Stichproben NeinJa Mehrfachvergleiche: Post hoc tests Mehrfachvergleiche: Bonferroni Korrektur, Šidàk-Bonferonni correction

2 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Data analysisData mining ReductionClassification Data Relationships Principal Component Analysis Factor Analysis Correspondence Analysis Homogeneity Analysis Non-linear PCA Procrustes Analysis Factor Analysis Discriminant Analysis Hierarchical Cluster Analysis Multidimensional Scaling K-Means Artificial Neural Networks Multiple Regression Principal Component Regression Linear Mixture Analysis Partial Least Squares - 2 Partial Least Squares -1 Canonical Analysis Support Vector Machines ANN SVM ANN SVM Categorization of multivariate methods

3 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Vorgehen beim statistischen testen: a)Aufstellen der H0/H1-Hypothese b)Ein- oder zweiseitige Fragestellung c)Auswahl des Testverfahrens d)Festlegen des Signikanzniveaus (Fehler 1. und 2. Art) e)Testen f)Interpretation

4 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler In Population gilt Entscheidung aufgrund der Stichprobe H0H1 H1H0 richtig, mit 1-αβ-Fehler P(H0¦H1)= β α-Fehler P(H1¦H0)= α richtig, mit 1- β Fehler 1. und 2. Art

5 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Bestimmen von Irrtumswahrscheinlichkeiten sei eine normalverteilte Stichprobe (nach 1. Grenzwertsatz) unbekannter Herkunft, mit Probe stammt aus dem Hunsrück Probe stammt aus der Eifel

6 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Test: Einstichproben Gauss Test mit Wert schneidet 0.62% von NV ab (P-Wert = Irrtumswahrscheinlichkeit) H0 muss verworfen werden! P-Wert wird gleiner mit > Diff. mit < mit > n α=5%, ~Z=1.65

7 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Frage: Welches muss überschritten werden, um H0 mit gerade verwerfen zu können? schneided von der rechten Seite der SNV genau 5% ab

8 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Zweiseitiger Test: schneidet auf jeder Seite der SNV genau 2.5% ab H0 wird knapper abgelehnt! Entscheidung ein-/zweiseitiger Test muss im Vorfeld erfolgen!

9 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Der β-Fehler Kann nur bei spezifischer H1 bestimmt werden! Wir testen, ob sich die Stichprobe mit dem Parameter der Eifelproben verträgt Wert schneidet auf der linken Seite der SNV 10.6% ab. Entscheidet man sich aufgrund des Ereignisses für die H0, so wird man mit einer p von 10.6% einen β- Fehler begehen, d.h. H1 (« Probe stammt aus der Eifel ») verwerfen, obwohl sie richtig ist.

10 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Die Teststärke Die β-Fehlerwahrscheinlichkeit gibt an, mit welcher p die H1 verworfen wird, obwohl ein Unterschied besteht 1- β gibt die p an zugunsten von H1 zu entscheiden, wenn H1 gilt. Bestimmen der Teststärke Wir habe herausgefunden, dass ab einem Wert der Test gerade signifikant wird (« Probe stammt aus der Eifel »)

11 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Bestimmen der Teststärke β-Wahrscheinlichkeit: Teststärke: 1-β = = Die p, dass wir uns aufgrund des gewählten Signifikanzniveaus (α=5%) zu Recht zugunsten der H1 entscheiden, beträgt 98.21% Determinanten der Teststärke: Mit kleiner werdener Diff. µ0-µ1 verringert sich 1- β Mit wachsendem n vergrössert sich 1- β Mit wachsender Merkmalsstreuung sinkt 1- β

12 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Why multivariate statistics? Fancy statistics do not make up for poor planning Design is more important than analysis Remember

13 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Prediction Methods –Use some variables to predict unknown or future values of other variables. Description Methods –Find human-interpretable patterns that describe the data. From [Fayyad, et.al.] Advances in Knowledge Discovery and Data Mining, 1996 Categorization of multivariate methods

14 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Multiple Linear Regression Analysis The General Linear Model A general linear model can be: straight-line model quadratic model (second-order model) more than one independent variables. E.g. x1x1 x2x2 y 0 =10 0 (x i1, x i2 ) E(yi)E(yi) yiyi i Response Surface

15 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler y=x 1 + x 2 – x x x 2 2 Multiple Linear Regression Analysis

16 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler The goal of an estimator is to provide an estimate of a particular statistic based on the data. There are several ways to characterize estimators: Bias: an unbiased estimator converges to the true value with large enough sample size. Each parameter is neither consistently over or under estimated Likelihood: the maximum likelihood (ML) estimator is the one that makes the observed data most likely ML estimators are not always unbiased for small N Efficient: an estimator with lower variance is more efficient, in the sense that it is likely to be closer to the true value over samples the best estimator is the one with minimum variance of all estimators Parameter Estimation Multiple Linear Regression Analysis

17 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler A linear model can be written as Where : is an N-dimensional column vector of observations is a (k+1)-dimensional column vector of unknown parameters is an N-dimensional random column vector of unobserved errors Matrix X is written as The first column of X is the vector, so that the first coefficient is the intercept. The unknown coefficient vector is estimated by minimizing the residual sum of squares Multiple Linear Regression Analysis

18 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Mean of errors is zero: Errors have a constant variance: Errors from different observations are independent of each other: for Errors follow a Normal Distribution. Errors are not uncorrelated with explanatory variable: Model assumptions The OLS estimator can be considered as the best linear unbiased estimator (BLUE) of provided some basic assumptions regarding the error term are satisfied : Multiple Linear Regression Analysis

19 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler For a multiple regression model 1 should be interpreted as change in y when a unit change is observed in x 1 and x 2 is kept constant. This statement is not very clear when x 1 and x 2 are not independent. Misunderstanding: i always measures the effect of x i on E(y), independent of other x variables. Misunderstanding: a statistically significant value establishes a cause and effect relationship between x and y. Interpreting Multiple Regression Model X2 X1 Y Multiple Linear Regression Analysis

20 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler If the model is useful… At least one estimated must 0 But wait …What is the chance of having one estimated significant if I have 2 random x? For each, prob(b 0) = 0.05 At least one happen to be b 0, the chance is: Prob(b1 0 or b2 0) = 1 – prob(b1=0 and b2=0) = 1-(0.95) 2 = Implication? Explanation Power by Multiple Linear Regression Analysis

21 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler R 2 (multiple correlation squared) – variation in Y accounted for by the set of predictors Adjusted R 2. The adjustment takes into account the size of the sample and number of predictors to adjust the value to be a better estimate of the population value. Adjusted R 2. The adjustment takes into account the size of the sample and number of predictors to adjust the value to be a better estimate of the population value. Adjusted R 2 = R 2 - (k - 1) / (n - k) * (1 - R 2 ) Where: n = # of observations, k = # of independent variables, Accordingly: smaller n decreases R 2 value; larger n increases R 2 value; smaller k, increases R 2 value; larger k, decreases R 2 value. The F-test in the ANOVA table to judge whether the explanatory variables in the model adequately describe the outcome variable. The F-test in the ANOVA table to judge whether the explanatory variables in the model adequately describe the outcome variable. The t-test of each partial regression coefficient. Significant t indicates that the variable in question influences the Y response while controlling for other explanatory variables. The t-test of each partial regression coefficient. Significant t indicates that the variable in question influences the Y response while controlling for other explanatory variables. Analysis Multiple Linear Regression Analysis

22 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Source of VarianceSSdfMS Regressionp-1 MSR=SSR/(p-1) Errorn-p MSE=SSE/(n-p) Totaln-1 ANOVA where J is an n n matrix of 1s Multiple Linear Regression Analysis

23 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler The R 2 statistic measures the overall contribution of Xs. Then test hypothesis: H0: 1=… k=0 H1: at least one parameter is nonzero Since there is no probability distribution form for R 2, F statistic is used instead. Multiple Linear Regression Analysis

24 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler F-statistics Multiple Linear Regression Analysis

25 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler How many variables should be included in the model? Basic strategies: Sequential forward Sequential backward Force entire The first two strategies determine a suitable number of explanatory variables using the semi-partial correlation as criterion and a partial F-statistics which is calculated from the error terms from the restricted (RSS 1 ) and unrestricted (RSS) models: where k, k 1 denotes the number of lags of the unrestricted and restricted model, and N is the number of observations. Multiple Linear Regression Analysis

26 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Measures the relationship between a predictor and the outcome, controlling for the relationship between that predictor and any others already in the model. It measures the unique contribution of a predictor to explaining the variance of the outcome. The semi-partial correlation Z X Y Multiple Linear Regression Analysis

27 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler An unbiased estimator for the variance is The regression coefficients are tested for significance under the Null-Hypothesis using a standard t-test Where denotes the i th diagonal element of the matrix. is also referred to as standard error of a regression coefficient. Testing the regression coefficients Multiple Linear Regression Analysis

28 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Which X is contributing the most to the prediction of Y? Cannot interpret relative size of bs because each are relative to the variables scale but s (Betas; standardized Bs) can be interpreted. a is the mean on Y which is zero when Y is standardized Multiple Linear Regression Analysis

29 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Can the regression equation be generalized to other data? Can be evaluated by randomly separating a data set into two halves. Estimate regression equation with one half and apply it to the other half and see if it predicts Cross-validation Multiple Linear Regression Analysis

30 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Residual analysis Multiple Linear Regression Analysis

31 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Divide the residuals into two (or more) groups based the level of x, The variances and the means of the two groups are supposed to be equal. A standard t-test can be used to test the difference in mean. A large t indicates nonconsistancy. e x/E(y) 0 The Revised Levene s test Multiple Linear Regression Analysis

32 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Influential points are those whose exclusion will cause major change in fitted line. Leave-one-out crossvalidation. If e i > 4s, it is considered as outlier. True outlier should not be removed, but should be explained. Detecting Outliers and Influential Observations Multiple Linear Regression Analysis

33 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Example for a Generalized Least- Square model which can be used instead of OLS-regression in the case of autocorrelated error terms (e.g. in Distributed Lag-Models) Generalized Least-Squares Multiple Linear Regression Analysis

34 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler SPSS-Example Multiple Linear Regression Analysis

35 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler SPSS-Example Multiple Linear Regression Analysis

36 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler SPSS-Example Model evaluation Multiple Linear Regression Analysis

37 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Studying residual helps to detect if: Model is nonlinear in function Missing x One or more assumptions of is violated. Outliers SPSS-Example Model evaluation Multiple Linear Regression Analysis

38 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler ANalysis Of VAriance ANOVA (ONE-WAY) ANOVA (TWO-WAY) MANOVA ANOVA

39 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Comparing more than two groups ANOVA deals with situations with one observation per object, and three or more groups of objects The most important question is as usual: Do the numbers in the groups come from the same population, or from different populations? ANOVA

40 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler One-way ANOVA: Example Assume treatment results from 13 soil plots from three different regions: –Region A: 24,26,31,27 –Region B: 29,31,30,36,33 –Region C: 29,27,34,26 H 0 : The treatment results are from the same population of results H 1 : They are from different populations ANOVA

41 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Comparing the groups Averages within groups: –Region A: 27 –Region B: 31.8 –Region C: 29 Total average: Variance around the mean matters for comparison. We must compare the variance within the groups to the variance between the group means. ANOVA

42 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Variance within and between groups Sum of squares within groups: Sum of squares between groups: The number of observations and sizes of groups has to be taken into account! ANOVA

43 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Adjusting for group sizes Both are estimates of population variance of error under H 0 n: number of observations K: number of groups If populations are normal, with the same variance, then we can show that under the null hypothesis, Reject at confidence level if ANOVA

44 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Continuing example -> H0 can not be rejected ANOVA

45 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler ANOVA table Source of variation Sum of squares Deg. of freedom Mean squares F ratio Between groups SSGK-1MSG Within groups SSWn-KMSW TotalSSTn-1 NOTE:

46 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler When to use which method In situations where we have one observation per object, and want to compare two or more groups: –Use non-parametric tests if you have enough data For two groups: Mann-Whitney U-test (Wilcoxon rank sum) For three or more groups use Kruskal-Wallis –If data analysis indicate assumption of normally distributed independent errors is OK For two groups use t-test (equal or unequal variances assumed) For three or more groups use ANOVA ANOVA

47 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Two-way ANOVA (without interaction) In two-way ANOVA, data fall into categories in two different ways: Each observation can be placed in a table. Example: Both type of fertilization and crop type should influence soil properties. Sometimes we are interested in studying both categories, sometimes the second category is used only to reduce unexplained variance. Then it is called a blocking variable ANOVA

48 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Sums of squares for two-way ANOVA Assume K categories, H blocks, and assume one observation x ij for each category i and each block j block, so we have n=KH observations. –Mean for category i: –Mean for block j: –Overall mean: ANOVA

49 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Sums of squares for two-way ANOVA ANOVA

50 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler ANOVA table for two-way data Source of variation Sums of squares Deg. of freedom Mean squaresF ratio Between groupsSSGK-1MSG= SSG/(K-1)MSG/MSE Between blocksSSBH-1MSB= SSB/(H-1)MSB/MSE ErrorSSE(K-1)(H-1)MSE= SSE/(K-1)(H-1) TotalSSTn-1 Test for between groups effect: compare to Test for between blocks effect: compare to

51 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Two-way ANOVA (with interaction) The setup above assumes that the blocking variable influences outcomes in the same way in all categories (and vice versa) Checking interaction between the blocking variable and the categories by extending the model with an interaction term ANOVA

52 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Sums of squares for two-way ANOVA (with interaction) Assume K categories, H blocks, and assume L observations x ij1, x ij2, …,x ijL for each category i and each block j block, so we have n=KHL observations. –Mean for category i: –Mean for block j: –Mean for cell ij: –Overall mean: ANOVA

53 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Sums of squares for two-way ANOVA (with interaction) ANOVA

54 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler ANOVA table for two-way data (with interaction) Source of variationSums of squares Deg. of freedom Mean squaresF ratio Between groupsSSGK-1MSG= SSG/(K-1)MSG/MSE Between blocksSSBH-1MSB= SSB/(H-1)MSB/MSE InteractionSSI(K-1)(H-1)MSI= SSI/(K-1)(H-1) MSI/MSE ErrorSSEKH(L-1)MSE= SSE/KH(L-1) TotalSSTn-1 Test for interaction: compare MSI/MSE with Test for block effect: compare MSB/MSE with Test for group effect: compare MSG/MSE with

55 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Notes on ANOVA All analysis of variance (ANOVA) methods are based on the assumptions of normally distributed and independent errors The same problems can be described using the regression framework. We get exactly the same tests and results! There are many extensions beyond those mentioned ANOVA

56 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA Uses Multiple DVs Various measures of soil properties –Corg, Cmik, N, pH,… Various outcome measures following different types of categories –Fertilization, point in time, crop type,… Predictors (IVs)Criterion (DV(s)) ANOVAMultiple, discreteSingle, continuous MANOVAMultiple, discreteMultiple, continuous MANOVA

57 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Multiple DVs could be analysed using multiple ANOVAs, but: –The FW increases with each ANOVA –Scores on the DVs are likely correlated Non-independent, and taken from the same subjects Hard to interpret results if multiple ANOVAs are significant MANOVA solves this by conducting only one overall test –Creates a composite DV –Tests for significance of the composite DV MANOVA

58 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler The Composite DV is a linear combination of the DVs –i.e., a discriminant function, or root –The weights maximally separate the groups on the composite DV C = W 1 Y 1 + W 2 Y 2 + W 3 Y 3 + …+ W n Y n where, C is a subjects score on the composite DV Y i are scores on each of the DVs W i are the weights, one for each DV A composite DV is required for each main effect and interaction MANOVA

59 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Considering the DVs together can enhance power a.Frequency distributions show considerable overlap between groups on the individual DVs b.The elipses, that reflect the DVs in combination, show less overlap c.Small differences on each DV combine to make a larger multivariate difference MANOVA

60 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler In ANOVA, the sums of squared deviations are partitioned: SS T = SS A + SS B + SS AxB + SS S/AB In MANOVA, the sum of squares cross-products are partitioned: S T = S D + S Tr + S DxTr + S S(DTr) The SSCP matrices (S) are analogous to the SS –SSCP matrix is a squared deviation that also reflects correlations among the DVs MANOVA

61 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Scores and Means in MANOVA are Vectors Y: Scores for each subject T and D: Row and column marginals GM: the grand mean DTr: the average scores of subjects within cells MANOVA

62 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA

63 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler The deviation score for the first subject is: The squared deviation is obtained by multiplying by the transpose: SS are on the diagonal: (25.89) 2 = 670, and (20.78) 2 = 431 Cross-products are on the off-diagonals: (25.89)(20.78)=538 And: MANOVA

64 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler The squaring of a matrix is carried out by multiplying it by its transpose The transpose is obtained by flipping the matrix about its diagonal: To multiply, the ij th element in the resulting matrix is obtained by the sum of products of the i th row in A and the j th column in A' For a vector, the transpose is a row vector, and: MANOVA

65 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Main Effects in ANOVA vs. MANOVA: The Interaction: The Error Term: MANOVA

66 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler In ANOVA, variance estimates (MS) are obtained from the SS for significance testing using the F-statistic In MANOVA, variance estimates (determinants) are obtained from the SSCP matrices for significance testing e.g. using Wilks Lambda ( ) ANOVAMANOVA SS~ SSCP MS~ |SSCP| ~ Note that F and are inverse to one another MANOVA

67 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler The determinant of a 2x2 matrix is given by: The determinants required to test the interaction are: Wilks Lambda for the Interaction is obtained by: MANOVA

68 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler If the effect is small, then approaches 1.0 –Here S DT was small, and was 0.91 Eta Squared for MANOVA is: 2 = 1 - Effect = 1 – 0.91 = 0.09 The interaction accounts for only 9% of the variance in the group means on the composite DV MANOVA

69 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Must have more cases/cell than number of DVs –Avoids singularity, enhances power Linear relation of all DVs and of DVs and COVs Multivariate normality –Sampling distribution of means for all DVs and linear combinations of DVs is normal Homogeneity of Variance-Covariance matrices –To rationalize pooling of error estimate Can be extended to within-subjects and mixed designs –Repeated measures are treated as new DVs MANOVA

70 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA SPSS Example MANOVA

71 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA SPSS Example MANOVA

72 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA

73 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA

74 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA

75 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Discriminant analysis is used to predict group memberships from a set of continuous predictors Analogy to MANOVA: in MANOVA linearly combined DVs are created to answer the question if groups can be separated. The same DVs can be used to predict group membership!! Discriminant Analysis

76 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler What is the goal of Discriminant Analysis? Perform dimensionality reduction while preserving as much of the class discriminatory information as possible. Seeks to find directions along which the classes are best separated. Takes into consideration the scatter within-classes but also the scatter between-classes. Discriminant Analysis

77 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler MANOVA and Disriminant Analysis (DA) are mathematically identical but are different in terms of emphasis: –DA is usually concerned with grouping of objects (classification) and testing how well objects were classified (one grouping variable, one or more predictor variables) –Discriminant functions are identical to canonical correlations between the groups on one side and the predictors on the other side. –MANOVA is applied to test if groups significantly differ from each other (one or more grouping variables, one or more predictor variables) Discriminant Analysis

78 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Discriminant Analysis

79 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Assumptions –small number of samples might lead to overfitting. –If there are more DVs than objects in any cell the cell will become singular and cannot be inverted. –If only a few cases more than DVs equality of covariance matrices is likely to be rejected. –With a small objects/DV ratio power is likely to be very small –Multivariate normality: the means of the various DVs in each cell and all linear combinations of them are normally distributed –Absence of outliers – significance assessment is very sensitive to outlying cases –Homogeneity of Covariance Matrices. DA is relatively robust to violations of this assumption if interference is the focus of the analysis, but not in classification. Discriminant Analysis

80 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Assumptions For classification purposes DA is highly influenced by violations for the last assumption, since subjects will tend to be classified into groups with the largest variance Homogeneity of class variances can be assessed by plotting pairwise the discriminant function scores for the first discriminant functions. LDA assumes linear relationships between all predictors within each group. Violations tend to reduce power and not increase alpha. Absence of Multicollinearity/Singularity in each cell of the design: Avoid redundant predictors Discriminant Analysis

81 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Interpreting a Two-Group Discriminant Function In the two-group case, discriminant function analysis is analogous to multiple regression; the two-group discriminant analysis is also called Fisher linear discriminant analysis. In general, in the two-group case we fit a linear equation of the type: c = a + d 1 *x 1 + d 2 *x d m *x m where a is a constant and d 1 through d m are regression coefficients and c is the predicted class. The interpretation of the results of a two-group problem is straightforward and closely follows the logic of multiple regression: Those variables with the largest (standardized) regression coefficients are the ones that contribute most to the prediction of group membership. Discriminant Analysis

82 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Discriminant Functions for Multiple Groups When there are more than two groups, then we can estimate more than one discriminant function. For instance, when there are three groups, there exist a function for discriminating between group 1 and groups 2 and 3 combined, and another function for discriminating between group 2 and group 3. Canonical analysis. In a multiple group discriminant analysis, the first function is defined such that it provides the most overall discrimination between groups, the second provides second most, and so on. All functions are independent or orthogonal. Computationally, a canonical correlation analysis is performed that determines the successive functions and canonical roots. The number of function that can be calculated is: Min [number of groups-1;number of variables] Discriminant Analysis

83 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Eigenvalues Eigenvalus can be interpreted as the proportion of variance accounted for by the correlation between the respective canonical variates. Successive eigenvalues will be of smaller and smaller size. First, compute the weights that maximize the correlation of the two sum scores. After this first root has been extracted, you will find the weights that produce the second largest correlation between sum scores, subject to the constraint that the next set of sum scores does not correlate with the previous one, and so on. Canonical correlations. If the square root of the eigenvalues is taken, then the resulting numbers can be interpreted as correlation coefficients. Because the correlations pertain to the canonical variates, they are called canonical correlations. Discriminant Analysis

84 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Let be the total number of samples. And Suppose there are C classes Let µ i be the mean vector of class i, i = 1,2,…, C Within-class scatter matrix : Between-class scatter matrix: Where = mean of the entire data set and Discriminant Analysis

85 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Methodology –LDA computes a transformation that maximizes the between- class scatter while minimizing the within-class scatter: products of eigenvalues ! projection matrix : scatter matrices of the projected data y Discriminant Analysis

86 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Linear transformation implied by LDA –The linear transformation is given by a matrix U whose columns are the eigenvectors of the above problem. –The LDA solution is given by the eigenvectors of the generalized eigenvector problem: –Important: Since S b has at most rank C-1, the max number of eigenvectors with non-zero eigenvalues is C-1 (i.e., max dimensionality of sub-space is C-1) Discriminant Analysis

87 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler Does S w -1 always exist? –If S w is non-singular, we can obtain a conventional eigenvalue problem by writing: –In practice, S w is often singular when more variables than cases are involved in the analysis (M << N ) Discriminant Analysis

88 26/06/ Praktische Statistik für Umwelt- und Geowissenschaftler


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