Chapter 7 Analysis of ariance

Variation Inherent or Natural Variation Due to the cumulative effect of many small unavoidable causes. Also referred to as noise Special or Assignable Variation Due to a) improperly adjusted machine b) operator error c) defective raw material Data without dispersion information is false data. — Kaoru Ishikawa

Analysis Of Variance ANOVA is often used for studying the relationship between a response variable (Y) and one or more explanatory or predictor variables (X’s). The predictor variables are also called factors or treatments. While the response is quantitative, the predictors may be either quantitative or qualitative. However, quantitative predictors are analyzed as if they are qualitative (or categorical).

ANOVA — Application ANOVA can be used to Determine the statistical significance of effects To identify sources of variability in Y 。 To determine the signification of a regression equation. To determine which factors affect the output in a DOE.

ANOVA — Assumptions The observations are mutually independent. –Stat  Nonparametrics  Runs Test The k groups exhibit homogeneity of variance. i.e.  1 ² =  2 ² =  =  k ² –Stat  ANOVA  Test for Equal Variances The data from each of the k groups is normally distributed. i.e.Factor Level i ~ N (  i,  i ²) –Stat  Basic Statistics  Normality Test

ANOVA — Principle N(  1,  1 ²) N(  2,  2 ²) N(  3,  3 ²) N(  4,  4 ²) 组内 组间组间

ANOVA — Hypothesis Testing H 0 :  1 =  2 =  =  k all group means are equal H a :  i   j for some i  j at least one pair of group means is not equal ANOVA verifies the null hypothesis by comparing the variance between the groups against the variation within a group mean: The null hypothesis is rejected if

One-way Analysis of Variance 1. Satisfy level of measurement requirements –Dependent variable is interval (ordinal) –Independent variable designates groups 2. Satisfy assumption of normality –Skewness and kurtosis –Central Limit Theorem

One-way Analysis of Variance 3. Test assumption of equal variances among groups –Levene test of equality of variances 4. Make decision about null hypothesis based on –Probability of F-statistic <= alpha  reject null hypothesis –Probability of F-statistic > alpha  fail to reject null hypothesis

One-way Analysis of Variance 5. Draw conclusion about research hypothesis based on decision about null hypothesis –Reject null hypothesis  support research hypothesis –Fail to reject null hypothesis  do not support research hypothesis

One-Way ANOVA Treatment 1Treatment 2Treatment （ Minitab: H0 ： Data is Normal; Ha ： Data is NOT Normal ） 。 2. （ Minitab: H0 ：  1=  2 =  3 Ha ： at lease one is different ） 。.

– Stat  Basic Statistics  Normality Test H0 ： Data is Normal; Ha ： Data is NOT Normal P -Value

Stat  ANOVA  Test for Equal Variances H0 ：  1=  2 =  3 Ha ： at lease one is different

Welcome to Minitab, press F1 for help. One-way ANOVA: Hardness versus Treatment Analysis of Variance for Hardness Source DF SS MS F P Treatmen Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev Treatmen ( * ) Treatmen ( * ) Treatmen ( * ) Pooled StDev = Within There is only a 1.2% chance ….

Normal probability plot - indicates whether the data are normally distributed, other variables are influencing the response, or outliers exist in the data 。

Histogram - indicates whether the data are skewed or outliers exist in the data

Residuals versus fitted values - indicates whether the variance is constant, a nonlinear relationship exists, or outliers exist in the data

Residuals versus order of the data - indicates whether there are systematic effects in the data due to time or data collection order

Thanks for Your Attention