Anova lecture notes pdf

Discussion 1 Jan. HW Tips. Topic 1: Introduction to the principles of experimental design. Reading: [ Word ] [ PDF ]. Lab 1 Word PDF. Topic 2: Distributions, hypothesis testing, and sample size determination. Discussion 2 Jan. Lab 2 Word PDF. Lab 2 R Word T2a b. Topic 4: Orthogonal Contrasts. Discussion 3 Jan. Topic 5: Means separations Due: HW 3. Lab 3 Word PDF. Lab 3 R Word T3a b c. A short summary of this paper. For example: y Fish may be collected from three different regions of a lake, in order to compare their weights over the three locations.

For example: y Twenty plots of carrots are grown in a field. Each plot is randomly allocated to one of five fertilizers, with four plots for each fertilizer. At the end of the experiment, the carrots from each plot are weighed. The yield of carrots with different fertilizers is being studied.

After three weeks of teaching, each child is tested for understanding of the material taught. The different teaching methods are being compared. The drugs are being compared for their influence on the progress of the disease.

The goal of a study is to find out the relationships between certain explanatory factors and response variables. The nature of the study matters when it comes to interpretation of results. In general, external evidence is required to rule out possible alternative explanations for a cause-and-effect relationship. The method of analysis depends on the nature of the data and the purpose of the study.

It extends the two-sample t -test to compare the means from more than two groups. If there is a difference, determine the nature of the difference. Basic Concepts y We shall start with a simple real life problem that many of us face. Most of the gas users are customers of gas companies. Mensah, who buys her gas from ABC gas agent, has faced a problem in the recent past. She knew that she is supposed to get Mensah was living. Is it possible to test from this data whether the mean amount of gas per cylinder differs from agent to agent?

But there is a better statistical procedure to do this simultaneously. We shall see how this can be done. Source of Variation y You know that variation is inevitable in almost all the variables measurable characteristics that we come across in practice. You will agree that some of the possible reasons for this variation are one or more of the following:- 9 The gas refilling machine at the company does not fill every cylinder with exactly same amount of gas.

In other words, we are interested in one factor or, one-way analysis of variance. When the data are classified only with respect to one type of source of variation, we say that we have one-way classification data.

In many situations, one conducts experiments to study the effect of a single factor on a variable under study. Such experiments, known as one-factor experiments, lead to one-way classification data. Classification of Data The process of arranging data into homogenous group or classes according to some common characteristics present in the data is called classification. For Example: The process of sorting letters in a post office, the letters are classified according to the cities and further arranged according to streets.

In the one-way ANOVA example, we are modeling crop yield as a function of the type of fertilizer used. First we will use aov to run the model, then we will use summary to print the summary of the model. All of the variation that is not explained by the independent variables is called residual variance.

In the two-way ANOVA example, we are modeling crop yield as a function of type of fertilizer and planting density. First we use aov to run the model, then we use summary to print the summary of the model. Adding planting density to the model seems to have made the model better: it reduced the residual variance the residual sum of squares went from Sometimes you have reason to think that two of your independent variables have an interaction effect rather than an additive effect.

To test whether two variables have an interaction effect in ANOVA, simply use an asterisk instead of a plus-sign in the model:. If you have grouped your experimental treatments in some way, or if you have a confounding variable that might affect the relationship you are interested in testing, you should include that element in the model as a blocking variable. How do you decide which one to use? The Akaike information criterion AIC is a good test for model fit.

AIC calculates the information value of each model by balancing the variation explained against the number of parameters used. In AIC model selection, we compare the information value of each model and choose the one with the lowest AIC value a lower number means more information explained! From these results, it appears that the two. To check whether the model fits the assumption of homoscedasticity, look at the model diagnostic plots in R using the plot function:.

The diagnostic plots show the unexplained variance residuals across the range of the observed data. The normal Q-Q plot plots a regression between the theoretical residuals of a perfectly-homoscedastic model and the actual residuals of your model, so the closer to a slope of 1 this is the better. This Q-Q plot is very close, with only a bit of deviation. ANOVA tells us if there are differences among group means, but not what the differences are. There is also a significant difference between the two different levels of planting density.

From the ANOVA test we know that both planting density and fertilizer type are significant variables. We can make three labels for our graph: A representing , B representing all the intermediate combinations , and C representing