Problems 6.15, 6.22
Q6.15. a nickel-titanium alloy is used to make components for jet turbine aircraft engines. Cracking is a potentially serious problem in the final part because it can lead to non-recoverable failures. A test is run at the parts producer to determine the effect of four factors on the cracks. The four factors are: Pouring temp (A), Titanium content (B), Heat treatment method (C), and amount of grain refiner used (D). Two replicates of a 2^4 design are run and the length of cracks (in mm x10^-2) induced in a sample coupon subjected to a standard test. a) Estimate the factor effects, which factor effects appear to be large.

(Half normal plot showing the significant factors)

From the minitab output given above after conducting an analysis of factorial design and showing the half normal plot it appear that the Pouring temp (A), Titanium content (B), Heat treatment method (C), and amount of grain refiner used (D) are all large and significant as well as the interaction between Pouring temp (A) and Titanium content (B), and interaction between Titanium content (B) and Heat treatment method (C), and finally the interaction between Pouring temp (A), Titanium content (B) and Heat treatment method (C). A, B, C, D, A&B, B&C, and A&B&C

b) Conduct an analysis of variance. Do any of the factors affect the cracking? Use α = 0.05.

From the analysis of variance minitab output given above we can conclude that the following factors affect the cracking for having a p-value < α=0.05: * Factor (A) Pouring temperature
* Factor (B) Titanium content
* Factor (C) Heat treatment method
* Factor (D) Grain refiner
* Interaction of (A) and (B)
* Interaction of (B) and (C)
* Interaction of (A), (B) and (C)

c) Write down the regression model that can be used to predict crack length as a function of the significant main effect and interaction you have identified in part (b).

...DESIGN AND ANALYSIS OF EXPERIMENTS
TERM PROJECT PROPOSAL
Subject: Statistical analysis of a sling regarding three factors with three levels.
Aim: Our aim is to statistically analyse the effects of three factors; rubber type, shooting range and tensile distance on the shooting range.
Description: In our project, we will design three slings for three types of rubbers.With these slings, we will try three shooting angles;30, 45, 60 degrees. Also with these factors we will make experiments with three tensile distances, namely the distance that we will pull the rubber; 2, 4 and 6 cm. As response values we will use the range that particular object goes. We will use the same pebble. So there will be no difference in the trials with respect to the used object.
Thus, in the analysis, we will examine the effects of rubber, angle and the distance on the range of the object takes after being released from the sling.
At the end, we will use Design Expert software for ANOVA and interpretations from the related graphs for concluding remarks from the experiment regarding the factors.
[pic]
PROJECT ANALYSIS
As we defined in the outline, we evaluated the effects of three factors such as; rubber type, angle and tensile for the shooting range of a sling. From our experiments we got 81 response values with different levels of the factors.
The structure of the experiment and data can be...

...LC•GC Europe Online Supplement
statistics and data analysis
9
Analysis of Variance
Shaun Burke, RHM Technology Ltd, High Wycombe, Buckinghamshire, UK. Statistical methods can be powerful tools for unlocking the information contained in analytical data. This second part in our statistics refresher series looks at one of the most frequently used of these tools: Analysis of Variance (ANOVA). In the previous paper we examined the initial steps in describing the structure of the data and explained a number of alternative significance tests (1). In particular, we showed that t-tests can be used to compare the results from two analytical methods or chemical processes. In this article, we will expand on the theme of significance testing by showing how ANOVA can be used to compare the results from more than two sets of data at the same time, and how it is particularly useful in analysing data from designed experiments.
With the advent of built-in spreadsheet functions and affordable dedicated statistical software packages, Analysis of Variance (ANOVA) has become relatively simple to carry out. This article will therefore concentrate on how to select the correct variant of the ANOVA method, the advantages of ANOVA, how to interpret the results and how to avoid some of the pitfalls. For those wanting more detailed theory than is given in the following section, several texts...

...Chapter 11 Two-Way ANOVA
An analysis method for a quantitative outcome and two categorical explanatory variables.
If an experiment has a quantitative outcome and two categorical explanatory variables that are deﬁned in such a way that each experimental unit (subject) can be exposed to any combination of one level of one explanatory variable and one level of the other explanatory variable, then the most common analysis method is two-way ANOVA. Because there are two diﬀerent explanatory variables the eﬀects on the outcome of a change in one variable may either not depend on the level of the other variable (additive model) or it may depend on the level of the other variable (interaction model). One common naming convention for a model incorporating a k-level categorical explanatory variable and an m-level categorical explanatory variable is “k by m ANOVA” or “k x m ANOVA”. ANOVA with more that two explanatory variables is often called multi-way ANOVA. If a quantitative explanatory variable is also included, that variable is usually called a covariate. In two-way ANOVA, the error model is the usual one of Normal distribution with equal variance for all subjects that share levels of both (all) of the explanatory variables. Again, we will call that common variance σ 2 . And we assume independent errors.
267
268
CHAPTER 11. TWO-WAY ANOVA
Two-way (or multi-way) ANOVA is an appropriate analysis...

...Analysis of Variance
Lecture 11 April 26th, 2011
A. Introduction
When you have more than two groups, a t-test (or the nonparametric equivalent) is no longer applicable. Instead, we use a technique called analysis of variance. This chapter covers analysis of variance designs with one or more independent variables, as well as more advanced topics such as interpreting significant interactions, and unbalanced designs.
B. One-Way Analysis of Variance
The method used today for comparisons of three or more groups is called analysis of variance (ANOVA). This method has the advantage of testing whether there are any differences between the groups with a single probability associated with the test. The hypothesis tested is that all groups have the same mean. Before we present an example, notice that there are several assumptions that should be met before an analysis of variance is used.
Essentially, we must have independence between groups (unless a repeated measures design is used); the sampling distributions of sample means must be normally distributed; and the groups should come from populations with equal variances (called homogeneity of variance).
Example:
15 Subjects in three treatment groups X,Y and Z.
X Y Z
700 480 500
850 460...

...of the MANOVA, check outcomes that test other assumptions for this statistic: equality of covariance matrices (see Box's Test) and sufficient correlation among the DVs (see Bartlett's Test of Sphericity). Also check the results of the Levene's Test of Equality of Error Variances to evaluate that assumption for the univariate ANOVAs that are run and show in the Tests of Between-Subjects Effects output. What have you found about whether the data meet these additional assumptions for the MANOVA and follow-up ANOVAs? Explain.
HINTS:
Once in the Options box, remember to check box for "Residual SSCP matrix" to get results for the Bartlett's test.
Also, remember to ask for post hoc tests for Treatment because there are more than two conditions. Profile plots also help with visualizing interactions.
6. What are the outcomes of the multivariate tests (main effects and interaction)? Report either the Pillai's Trace or Wilks's Lambda for each result, as well as the associated F-value and its statistical significance. Use the following format for notation to report each result: Pillai's Trace OR Wilks' lambda = ____; F(df, df) = ____, p = ____.
HINTS:
Use Pillai's trace if there are problems with heterogeneity of variance-covariance matrices for the DVs. Otherwise, Wilks' lambda is fine.
Eta squared cannot be calculated from the information provided in the multivariate tests results.
7. Given the results of the multivariate tests, would you now move...

...INTRODUCTION TO ONE-WAY ANALYSIS OF VARIANCE
Dale Berger, Claremont Graduate University http://wise.cgu.edu
The purpose of this paper is to explain the logic and vocabulary of one-way analysis of variance (ANOVA). The null hypothesis tested by one-way ANOVA is that two or more population means are equal. The question is whether (H0) the population means may equal for all groups and that the observed differences in sample means are due to random sampling variation, or (Ha) the observed differences between sample means are due to actual differences in the population means.
The logic used in ANOVA to compare means of multiple groups is similar to that used with the t-test to compare means of two independent groups. When one-way ANOVA is applied to the special case of two groups, one-way ANOVA gives identical results as the t-test.
Not surprisingly, the assumptions needed for the t-test are also needed for ANOVA. We need to assume:
1) random, independent sampling from the k populations;
2) normal population distributions;
3) equal variances within the k populations.
Assumption 1 is crucial for any inferential statistic. As with the t-test, Assumptions 2 and 3 can be relaxed when large samples are used, and Assumption 3 can be relaxed when the sample sizes are roughly the same for each group even for small samples. (If there are extreme outliers or errors in...

... 2/21/2014
274 EXERCISE 36 • Analysis of Variance (ANOVA) I
1. The researchers found a significant difference between the two groups (control and treatment) for change
in mobility of the women with osteoarthritis (OA) over 12 weeks with the results of F(1, 22) 9.619,
p 0.005. Discuss each aspect of these results. F is the statistic for ANOVA. F (1,22) one represents the number of groups in the study and 22 equals the subjects used the error df, and 9.619 is significant as it is P=0.005, it can be said that the intervention group participants face a significant reduction in mobility difficulty.
2. State the null hypothesis for the Baird and Sands (2004) study that focuses on the effect of the GI with
PMR treatment on patients’ mobility level. Should the null hypothesis be rejected for the difference between
the two groups in change in mobility scores over 12 weeks? Provide a rationale for your answer.
Ho 1: Guided imagery (GI) with Progressive Muscle Relaxation reduces pain difficulties of women with OA.
Ho 2: Guided imagery (GI) with Progressive Muscle Relaxation reduces mobility difficulties of women with OA.
The null hypothesis should be accepted study results indicate a significant improvement in mobility and pain difficulties.
3. The researchers stated that the participants in the intervention group reported a reduction in mobility
difficulty at week 12. Was this result statistically significant,...

...
1. Industrial situations that show the usefulness of analysis of covariance
Analysis of covariance (ANCOVA) gives evaluation of whether the population means on the dependent variable (DV) adjusted for differences on the covariate(s), are different across the independent variable (IV) levels. The variability in the DV due to the control variable (concomitant variable or a covariate) is removed.
The covariate increases the opportunity to find statistical significance for the factors, fixed or random. Simply put, the covariate independent variable reduces the variance in the dependent variable that is to be explained by the factors.
An example of ANCOVA is testing the effect of computer experience on the use of e-shopping, making attitude towards e-shopping as a covariate. The covariate removes its effect from the e-shopping measure, since in general those with positive attitudes shop more.
2. Real life industrial research questions that could be analyzed by partial correlation
Below are some of the research questions that can be analyzed by partial correlation:
What is the effect of job satisfaction on salary, type of education, gender, and smoker, factoring out age?
Is there an affect of electrical appliances on contraceptive use, removing the effect of education?
Is there an effect on prep hours and SAT scores, factoring the effect of GPA?
3. Variables
From the research questions above, the first one...