5 Lec - للترجمه.pptx

ANALYSIS OF VARIANCE (ANOVA) ANOVA • Analysis of variance is the special case of regression with nominal independent variable. • Uses: o In comparisons of several population means. o To analyze the differences between the means of two or more groups or treatments. o To determine whether there are...

ANALYSIS OF VARIANCE (ANOVA) ANOVA • Analysis of variance is the special case of regression with nominal independent variable. • Uses: o In comparisons of several population means. o To analyze the differences between the means of two or more groups or treatments. o To determine whether there are any statistically significant differences between the means of different groups. • ANOVA compares the variation between group means to the variation within the groups. • If the variation between group means is significantly larger than the variation within groups, it suggests a significant difference between the means of the groups (Factor effect) Types of ANOVA • One-way (Our focus today) o evaluates the impact of a sole factor on a sole response variable. o determines whether all the samples are the same. o determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups (not variables). o Example: drug effect • Two-way there are two independent variables. o It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time. o Example: drug and gender effect Assumptions • Just like regression o Linearity o Independence o Normality o Equal variance Linearity check • Scatter plots is a common of determining if nonlinearity exists in a relationship. • Standardized residuals plot against standardized estimates. • When the standard deviation of the residuals exceeds the standard deviation of the dependent. • ANOVA test of linearity: o One can compute an ANOVA table for the linear and nonlinear components of any pair of variables. o If the F significance value for the nonlinear component is below the critical value (ex., <.05), then there is significant nonlinearity. Normality check • A normal distribution is assumed by many statistical procedures. • Various transformation methods are used to correct non-normally distributed data. • Normality can be visually assessed by o looking at a histogram of frequencies o looking at a normal probability plot output by most computer programs. o The area under the normal curve represents probability:  68.26% of cases will lie within 1 standard deviation of the mean  95.44% within 2 standard deviations  99.14% within 3 standard deviations. Normality check • Statistical methods o Skewness. o Kurtosis (tails to the middle of the distribution). o Shapiro-Wilks W test, Kolmogorov-Smirnov D test. • Graphical methods o Histogram o P-P plot (The straighter the line formed by the P-P plot, the more the variable's distribution conforms to the selected test normal). o Q-Q plot of the quantiles of a variable's distribution against the quantiles of the test distribution o Boxplot is a good tool for detecting outliers. Equal Variance (homogeneity of variance) • Levene's test of homogeneity of variance tests the assumption that each group (category) of the independent) (s) has the same variance on an interval dependent. • If the Levene statistic is significant at the .05 level or better, the researcher rejects the null hypothesis that the groups have equal variances. Random vs Fixed factors • Random Factor is a factor whose levels are sampled at random from a large population of levels, such as a selection of 3 drugs from a group on many available drugs. • Fixed Factors whose levels are not random sample, such as Gender, Religion, Race, Education, Marital status • Note that there are some cases we use some factors as fixed or random e.g. Locations, Treatments, Drugs, Tests Example • Five diets for hamsters were tested for differences in weight gain after a specified period of time. Five inbred laboratory lines were used to investigate the responses of different genotypes to the various diets. Are there significant differences among the diets in their ability to facilitate weight gain? The units are grams of increase.