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15-04-2011

Microarray Center. BIOSTATISTICS. Lecture 8 Inferences about Population Variance Analysis of Variance (ANOVA). dr. Petr Nazarov petr.nazarov@crp-sante.lu. 15-04-2011. OUTLINE. Lecture 8. Part I Interval estimation for population variance

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15-04-2011

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  1. Microarray Center BIOSTATISTICS Lecture 8 Inferences about Population VarianceAnalysis of Variance (ANOVA) dr. Petr Nazarovpetr.nazarov@crp-sante.lu 15-04-2011

  2. OUTLINE Lecture 8 • Part I • Interval estimation for population variance • variance sampling distribution, 2 statistics • calculation of interval estimation • hypothesis tests for a population variance • Comparison of variances of two populations • F-statistics • formulation of hypotheses and testing • Part II • Introduction to ANOVA • why ANOVA • shoe experiment • assumptions with ANOVA • Single-factor ANOVA • Multi-factor ANOVA • Experimental design • randomized design • block design

  3. LECTURE 8 Part II Part I Inference about Population Variance

  4. INTERVAL ESTIMATION FOR VARIANCE Variance Sampling Distribution Variance A measure of variability based on the squared deviations of the data values about the mean. population sample The interval estimation for variance is build using the following measure: Sampling distribution of (n-1)s2/2 Whenever a simple random sample of size n is selected from a normal population, the sampling distribution of (n-1)s2/2 has achi-squaredistribution (2) with n-1 degrees of freedom.

  5. INTERVAL ESTIMATION FOR VARIANCE 2 Distribution 2 distribution works only for sampling from normal population

  6. INTERVAL ESTIMATION FOR VARIANCE 2 Probabilities in Table and Excel • In Excel use functions: • = CHIDIST(2,n-1) • = CHIINV(, n-1)

  7. INTERVAL ESTIMATION FOR VARIANCE 2 Distribution for Interval Estimation 2 distribution for d.f. = 19

  8. INTERVAL ESTIMATION FOR VARIANCE Interval Estimation Suppose sample of n = 36 coffee cans is selected and m = 2.92 and s = 0.18 lbm is observed. Provide 95% confidence interval for the standard deviation • = CHIINV(, n-1)

  9. INTERVAL ESTIMATION FOR VARIANCE Hypotheses about Population Variance H0: 2  const Ha: 2 > const H0: 2  const Ha: 2 < const H0: 2 = const Ha: 2  const

  10. VARIANCES OF TWO POPULATIONS Sampling Distribution In many statistical applications we need a comparison between variances of two populations. In fact well-known ANOVA-method is base on this comparison. The statistics is build for the following measure: Sampling distribution of s12/s22 when 12= 22 Whenever a independent simple random samples of size n1 and n2 are selected from two normal populations with equal variances, the sampling of s12/s22 has F-distribution with n1-1 degree of freedom for numerator and n2-1 for denominator. F-distribution for 20 d.f. in numerator and 20 d.f. in denominator • In Excel use functions: • = FTEST(data1,data2)

  11. VARIANCES OF TWO POPULATIONS Hypotheses about Variances of Two Populations H0: 12  22 Ha: 12 > 22 H0: 12 = 22 Ha: 12 22

  12. VARIANCES OF TWO POPULATIONS Example schoolbus.xls

  13. VARIANCES OF TWO POPULATIONS Example schoolbus.xls 1. Let us start from estimation of the variances for 2 data sets interval estimation (optionally) Milbank: s12= 48 Gulf Park: s22 = 20 Milbank: 12 48 (29.5 91.5) Gulf Park: 22 20 (10.9 47.9) 2. Let us calculate the F-statistics 3. … and p-value = 0.08 p-value =0.08 <  = 0.1 • In Excel use one of the functions: • = 2*FDIST(F,n1-1,n2-1) • = FTEST(data1,data2)

  14. LECTURE 8 Part II Part II Analysis of Variance (ANOVA)

  15. If we would use pairwise comparisons, what will be the probability of getting error? Number of comparisons: Probability of an error: 1–(0.95)10 = 0.4 INTRODUCTION TO ANOVA Why ANOVA? Means for more than 2 populations We have measurements for 5 conditions. Are the means for these conditions equal? Validation of the effects We assume that we have several factors affecting our data. Which factors are most significant? Which can be neglected? ANOVA example from Partek™ http://easylink.playstream.com/affymetrix/ambsymposium/partek_08.wvx

  16. INTRODUCTION TO ANOVA Example As part of a long-term study of individuals 65 years of age or older, sociologists and physicians at the Wentworth Medical Center in upstate New York investigated the relationship between geographic location and depression. A sample of 60 individuals, all in reasonably good health, was selected; 20 individuals were residents of Florida, 20 were residents of New York, and 20 were residents of North Carolina. Each of the individuals sampled was given a standardized test to measure depression. The data collected follow; higher test scores indicate higher levels of depression. Q: Is the depression level same in all 3 locations? depression.xls H0: 1= 2= 3 Ha:not all 3 means are equal

  17. m2 m3 m1 INTRODUCTION TO ANOVA Meaning H0: 1= 2= 3 Ha:not all 3 means are equal

  18. INTRODUCTION TO ANOVA Assumptions for ANOVA Assumptions for Analysis of Variance 1. For each population, the response variable is normally distributed 2. The variance of the respond variable, denoted as 2 is the same for all of the populations. 3. The observations must be independent.

  19. INTRODUCTION TO ANOVA Some Calculations Let’s estimate the variance of sampling distribution. If H0 is true, then all mi belong to the same distribution – this is called between-treatment estimate, works only at H0 At the same time, we can estimate the variance just by averaging out variances for each populations: – this is called within-treatment estimate Does between-treatment estimate and within-treatment estimate give variances of the same “population”?

  20. due to treatment Sum squares Mean squares, beetween2 Test of variance equality p-value for the treatment effect due to error Sum squares Mean squares, within2 SINGLE-FACTOR ANOVA Theory Means for treatments Variances treatments Total mean H0: 1= 2= …= k Ha:not all k means are equal

  21. SINGLE-FACTOR ANOVA The Main Equation SS due to treatment Total sum squares SS due to error Total variability of the data include variability due to treatment and variability due to error Partitioning The process of allocating the total sum of squares and degrees of freedom to the various components.

  22. SINGLE-FACTOR ANOVA Example m2 m3 m1

  23. SSTR SSE SINGLE-FACTOR ANOVA Example ANOVA table A table used to summarize the analysis of variance computations and results. It contains columns showing the source of variation, the sum of squares, the degrees of freedom, the mean square, and the F value(s). • In Excel use: • Tools  Data Analysis  ANOVA Single Factor depression.xls Let’s perform for dataset 1: “good health”

  24. MULTI-FACTOR ANOVA Factors and Treatments Factor Another word for the independent variable of interest. Factorial experiment An experimental design that allows statistical conclusions about two or more factors. Treatments Different levels of a factor. good health bad health Factor 1: Health depression.xls Florida New York North Carolina Factor 2: Location Depression =  + Health + Location + HealthLocation +  Interaction The effect produced when the levels of one factor interact with the levels of another factor in influencing the response variable. ANOVA example from Partek™

  25. MULTI-FACTOR ANOVA 2-factor ANOVA with r Replicates Replications The number of times each experimental condition is repeated in an experiment.

  26. MULTI-FACTOR ANOVA 2-factor ANOVA with r Replicates: Example depression.xls Factor 1: Health Factor 2: Location 1. Reorder the data into format understandable for Excel 2. Use Tools  Data Analysis  ANOVA: Two-factor with replicates

  27. MULTI-FACTOR ANOVA 2-factor ANOVA with r Replicates: Example

  28. MULTI-FACTOR ANOVA Example 2 salaries.xls Q:Which factors have significant effect on the salary Tools  Data Analysis  ANOVA: Two-factor with replicates

  29. EXPERIMENTAL DESIGN Experiments Aware of Batch Effect ! When designing your experiment always remember about various factors which can effect your data: batch effect, personal effect, lab effect... Day 1 ? T =+30C Day 2 T = +10C

  30. EXPERIMENTAL DESIGN Experiments Completely randomized design An experimental design in which the treatments are randomly assigned to the experimental units. We can nicely randomize: Day effect Batch effect

  31. Day 1 Day 2 EXPERIMENTAL DESIGN Experiments Blocking The process of using the same or similar experimental units for all treatments. The purpose of blocking is to remove a source of variation from the error term and hence provide a more powerful test for a difference in population or treatment means.

  32. EXPERIMENTAL DESIGN Experiments A good suggestion…  Block what you can block, randomize what you cannot, and try to avoid unnecessary factors

  33. ANOVA Optional Task mice.xls Q:Does mouse strain affect the weight? Show the effects of sex and strain using ANOVA

  34. ANOVA Optional Task mice.xls B.t.w., something is wrong…. Can you find a problem here? 

  35. QUESTIONS ? Thank you for your attention to be continued…

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