Stats

Chi-Square Test for Independence Assumptions

-Simple random sample -All estimated expected numbers 𝐸𝑖𝑗 ≥ 1. -No more than 20% of the estimated expected numbers 𝐸𝑖𝑗 < 5.

Interaction Plot

A plot that shows the means at each treatment combination, highlighting the factors effects and their behavior at all the combinations.

Observational study

A study based on data in which no manipulation of factors has been employed.

Factor

A variable whose levels are controlled by the experimenter. Experiments attempt to discover the effects that differences in factor levels may have on the responses of the experimental units.

Response

A variable whose values are compared across different treatments. In a randomized experiment, large response differences can be attributed to the effect of differences in treatment level.

Which of the following is not an assumption for ANOVA? <a> Independence <b> Equal Variance <c> Normal Population <d> All of the above are assumptions for ANOVA

All of the above are assumptions for ANOVA

Analysis of Variance (ANOVA)

An analysis method for testing equality of means across treatment groups.

Experiment

An experiment manipulates factor levels to create treatments, randomly assigns subjects to these treatment levels, and then compares the responses of the subject groups across treatment levels.

Prospective study

An observational study in which subjects are followed to observe future outcomes. Because no treatments are deliberately applied, a prospective study is not an experiment. Nevertheless, prospective studies typically focus on estimating differences among groups that might appear as the groups are followed during the course of the study.

Retrospective study

An observational study in which subjects are selected and then their previous conditions or behaviors are determined. Because retrospective studies are not based on random samples, they usually focus on estimating differences between groups or associations between variables.

Blind Blinding

Any individual associated with an experiment who is not aware of how subjects have been allocated to treatment groups is said to be blinded.

multiple comparisons

If we REJECT the null hypothesis of equal means, we often then want to investigate further and compare pairs of treatment group means to see if the corresponding population means differ. If we want to test several such pairs, we must adjust for performing several tests to keep the overall risk of a Type I error from growing too large. Such adjustments are called methods for multiple comparisons.

Experimental Units

Individuals on whom an experiment is performed. Usually called subjects or participants when they are human.

Bonferroni method

One of many methods for adjusting the margin of error to control the overall risk of making a Type I error when testing many pairwise differences between group means.

An estimate of the variation within the groups

Sum of squares due to Error (SSE)

An estimate of the variation between the groups

Sum of squares due to Treatments (SSTr)

Anova Table

The ANOVA table is convenient for showing the degrees of freedom, treatment mean square, error mean square, their ratio, F-statistic, and its P-value. There are usually other quantities of lesser interest included as well.

F-Distribution

The F-distribution is the sampling distribution of the F-statistic when the null hypothesis that the treatment means are equal is true. The F is the ratio of two estimates of variance (mean squares), which are equal when the null hypothesis is true. It has two degrees of freedom parameters, corresponding to the degrees of freedom for the mean squares in the numerator and denominator respectively.

F-statistic

The F-statistic for one-way ANOVA is the ratio MSTr/MSE. When the F-statistic is sufficiently large, we reject the null hypothesis that the group means are equal.

F-test

The F-test tests the null hypothesis that all the group means are equal against the one-sided alternative that they are not all equal. We reject the hypothesis of equal means if the F-statistic exceeds the critical value from the F-distribution corresponding to the specified significance level and degrees of freedom.

Control Group

The experimental units assigned to a baseline treatment level, typically either the default treatment, which is well understood, or a null, placebo treatment. Their responses provide a basis for comparison.

Treatment

The process, intervention, or other controlled circumstance applied to randomly assigned experimental units. Treatments are the different levels of a single factor or are made up of combinations of levels of two or more factors.

Level

The specific values that the experimenter chooses for a factor are called the levels of the factor.

Placebo Effect

The tendency of many human subjects (often 20% or more of experiment subjects) to show a response even when administered a placebo.

Double-blind, Single-blind

There are two classes of individuals who can affect the outcome of an experiment: those who could influence the results (subjects, treatment administrators, or technicians) those who evaluate the results (judges, treating physicians, etc.) When every individual in either of these classes is blinded, an experiment is said to be single-blind. When everyone in both classes is blinded, we call the experiment double-blind.

If we reject the null hypothesis of equal means, we often then want to investigate further and compare pairs of treatment group means to see if the corresponding population means differ. If we want to test several such pairs, we must adjust for performing several tests to keep the overall risk of a __________ from growing too large. Such adjustments are called methods for multiple comparisons. <a> Type I error <b> Type II error

Type I error

Confounded

When a factor is associated with another factor in such a way that their effects cannot be separated, we say that these two factors are confounded.

Blocking, Blocking Factor

When groups of experimental units are similar, it is often a good idea to gather them together into the same level of a factor. The factor is called a blocking factor and its levels are called blocks. By blocking we isolate the variability attributable to the differences between the blocks so that we can see the differences in the means due to the treatments more clearly.

Interaction

When the effects of the levels of one factor change depending on the level of the other factor, two factors are said to interact. When interaction terms are present, it is misleading to talk about the main effect of one factor because how large it depends on the level of the other factor.

Subjects or Participants

When the experimental units are people, they are usually referred to as Subjects or Participants

Control

When we limit the levels of a factor not explicitly part of the experiment design, we have controlled that factor. (By contrast, the factors we are testing are said to be manipulated.)

placebo

`A treatment that mimics the treatment to be studied, designed so that all groups think they are receiving the same treatment. Many subjects respond to such a treatment (a response known as a placebo effect). Only by comparing with a placebo can we be sure that the observed effect of a treatment is not due simply to the placebo effect.

Parallel line plots indicate

a lack of interaction

What is the alternative hypothesis for ANOVA?

at least one mean is different

If 𝑝-value > α (level of significance)

fail to reject

Line plots that cross indicate a significant

interaction

If 𝑝-value < α (level of significance)

reject 𝐻0 in favor of 𝐻1

When an experiment has a randomized block design,

the randomization occurs only within blocks

Principles of experimental design

• Control aspects of the experiment that we know may have an effect on the response, but that are not the factors being studied. • Randomize subjects to treatments to even out effects that we cannot control. • Replicate over as many subjects as possible. Results for a single subject are just anecdotes. • Block to reduce the effects of identifiable attributes of the subjects that cannot be controlled.

Assumptions for ONE Factor Anova

• Independence Assumption o Randomization Condition • Equal Variance Assumption o Similar Variance Condition • Normal Population Assumption o o NearlyNormalCondition

Designs

• Randomized block design: The randomization occurs only within blocks. • Completely randomized design: All experimental units have an equal chance of receiving any treatment. • Factorial design: Includes more than one factor in the same design and includes every combination of all the levels of each factor.

Replication design two factor anova hypothesis

𝐻0: No interaction exists between factors display panel and emergency condition. 𝐻1: An interaction does exist.

Hypothesis for Chi Squared

𝐻0: The two characteristics are independent, not associated. 𝐻1: The two characteristics are dependent.

TWO WAY ANOVA Hypothesis test

𝐻0: 𝜇1∙ =𝜇2∙ =⋯=𝜇𝑎∙ 𝐻1: At least two means differ