Submit your answers to the following questions using the ANOVA source table below. The table depicts a two-way ANOVA in which gender has two groups (male and female), marital status has three groups (married, single never married, divorced), and the means refer to happiness scores (n = 100): a. What is/are the independent variable(s)? What is/are the dependent variable(s)? The independent variables are gender and marital status. The dependent variable is the happiness. b. What would be an appropriate null hypothesis? Alternate hypothesis? Alternate hypothesis about gender can be that females will have greater happiness mean score than males. There is also an alternative hypothesis in marital status that females who are married would have lower happiness mean scores than males that are married. The Null hypothesis in both situations would be that differences would not exist. c. What are the degrees of freedom for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance? 1. A-1=There are 2 groups of gender; male and female, so it would be 2-1=1 as the degree of freedom. 2. B-1=There are 3 groups for marital status; married, single, and divorced, so it would be 3-1=2 as the degree of freedom. 3. (A-1)*(B-1) =That would be the answer from gender (1)* The answer from marital status (2) which would be 1*2=2 for the degree of freedom. 4. N-AB= That would be the total number of observations which is given as N=100 minus gender*marital status. This would be; 100-6=94 for the error. d. Calculate the mean square for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance. 5. Sum of squares due to effect A/degrees of freedom of effect A=____. This would be 68.15/1= 68.15 for the mean square. 6. Sum of squares due to effect...

...Submit your answers to the following questions using the ANOVA source table below. The table depicts a two-wayANOVA in which gender has two groups (male and female), marital status has three groups (married, single never married, divorced), and the means refer to happiness scores (n = 100):
What is/are the independent variable(s)? What is/are the dependent variable(s)?
What would be an appropriate null hypothesis? Alternate hypothesis?
What are the degrees of freedom for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance?
Calculate the mean square for 1) gender, 2) marital status, 3) interaction between gender and marital status, and 4) error or within variance.
Calculate the F ratio for 1) gender, 2) marital status, and 3) interaction between gender and marital status.
Identify the critical Fs at alpha = .05 for 1) gender, 2) marital status, and 3) interaction between gender and marital status.
If alpha is set at .05, what conclusions can you make?
Source Sum of Squares (degrees of freedom [df]) Mean Square Fobt. Fcrit.
Gender 68.15 ? ? ? ?
Marital Status 127.37 ? ? ? ?
Gender * Marital Status (A x B) 41.90 ? ? ? ?
Error (Within) 864.82 ? ? NA NA
Total 1102.24 99 NA NA NA
Please Note: The table that you see in the assignment has been slightly modified from the one presented in the module notes since it is...

...Two-WayANOVATwo-wayANOVA is similar to one-wayANOVA in all aspects except that in this case additional independent variable is introduced. Each independent variable includes two or more variants.
Take the Demonstration problem 11.3 on page number 437 of KEN BLACK
Feed the data as shown below:
And the values as:
1 1 3.47
1 2 3.43
1 3 3.44
1 4 3.46
1 5 3.46
1 6 3.44
2 1 3.40
2 2 3.41
2 3 3.41
2 4 3.45
2 5 3.40
2 6 3.43
3 1 3.38
3 2 3.42
3 3 3.43
3 4 3.40
3 5 3.39
3 6 3.42
4 1 3.32
4 2 3.35
4 3 3.36
4 4 3.30
4 5 3.39
4 6 3.39
5 1 3.50
5 2 3.44
5 3 3.45
5 4 3.45
5 5 3.48
5 6 3.49
Click Analyze General Linear Model Univariate… This will open Univariate dialogue box.
Choose Price and send it to Dependent Variable Box. Similarly, choose City and Brand to send them in Fixed Factor(s) Box.
Click Options Push Button to open its sub dialogue box.
Click Descriptive Statics, Estimates of effect size, Observed Power and Homogeneity tests check boxes in the display box and click continue.
Previous Dialogue Box will open, click ok to see the output....

...One-way repeated measure ANOVA
In a one-way repeated measures ANOVA design,each subject is exposed to two or more different conditions, or measured on the continuous scale onthree or more occasions. It can also be used to compare respondents’ responses to two or more questions or items. These questions, hiwever, must be meausred using the same scale.( Likert scale)
Example of research question: Is there a change in confidence scores over the three time periods?
What you need: One group of participants measured on the same scale on three different occasions or under three different conditions, or each person measured on three different questions or items ( using the same scale). This involves teo variables:
● one indepenedet variable ( categorical) ( e.g. Time 1/Time 2/Time 3)
● one dependent variable ( numerical) (e.g scores on the confidence in coping with statistics test). The scores on the test for each time point will appear in the data file in different columns.
What it does: This technique will tell you if there is a significant difference somewhere among the three sets of scores
Assumptions: Same as ANOVA
Non-parametric alternative: Friedman Test
Example
A group of students were invited to participate in an intervention designed to increase their confidence in their ability to do statistics. The confidence levels wer measyred before intervention (Time 1),...

...of Variance (ANOVA)
Indian Institute of Public Health Delhi
MSc CR 2013-15
Outline of the session
• Need for Analysis of Variance
• Concept behind one wayANOVA
• Example
• Non-parametric alternative
When dependent variable is continuous
Type of
Dependent
variable
Type of
Independent
variable
Number
of
Groups
Continuous
Categorical
More
than
two
Non-parametric (Wilcoxon sign
rank)
Paired t – test
Not normal
Non-parametric (Wilcoxon sign
rank)
Independent z or t – test
Not normal
Non-parametric (Wilcoxon rank
sum or Mann-Whitney U )
Not normal
Unrelated or
independent
Not normal
Normal
Two
z or one sample t – test
Normal
Related
Choice of Significance test
Normal
NA
Distribution of
dependent
variable
Normal
One
Related/
Dependent
One wayANOVA/linear
regression
Non-parametric (Kruskal Wallis)
Normal
Repeated ANOVA
Not normal
Non-parametric (Friedmans test)
Unrelated
Related
Background
• When you have more than two groups to compare,
you can apply t-test multiple times
• But this is not done, why???
• Probability of type I error increases
• This increases as the number of comparison
increases
• Analysis of variance (ANOVA) is one way of dealing
with this problem which tests for...

...Chapter 15: Introduction to the Design of Experimental and Observational Studies
The Models in Analysis of Variance(ANOVA) and in Regression are diﬀerent. In regression model, all the response and predictors are continuous (quantitative) variables. However, in ANOVA model, the response is continuous but the predictors are categorical (qualitative) variables. There are some concepts here. 1. Factor and factor level. A factor is a predictor (explanatory or independent) variable. A factor level is a particular form of the factor. Mostly, the level can not be compared. 2. Single-factor and multi-factor studies. Single factor study means there is only one factor in the study. Thus, the model only includes one response and one predictor. Multi-factor means there are more than one factor in the study. An important case in the multi-factor study is two-wayANOVA model. 3. Experimental and Observational studies. An experimental study means the level of all the factors can be totally controlled. An observational study means the level of the factors can not be controlled. If some of them can be controlled and some of them can not, them people treat it as an experimental studies and called the controlled factor signal factor and uncontrolled factor noise factor. 4. If a continuous variable is treated as a categorical variable, then it is also called a factor variable. 5. Treatment and block. The factor can be...

...Decker et al. (2001) showed that eHx can be splited into three parts: back-of-plate (eb), front-of-plate (ef), and crossing the hole (eh). Gawlik and Kutscher (2002) studied numerically and experimentally the wind heat loss from UTCs with sinusoidal corrugations. They used a numerical model to determine heat loss to the air stream over the plate as a function of wind speed, suction velocity and plate geometry. The test conditions used for experimental runs were used as inputs to the numerical model to determine whether it predicted convective heat loss accurately. Correlations for heat loss from the plate to the crosswind were determined for both the attached and separated ﬂow cases. Gawlik et al. (2005) compared the thermal performance of two plate geometries made with high and low conductivity material under several ﬂow conditions. They concluded that the eﬀect of plate conductivity on the thermal performance of a UTC is small and that low-conductivity material can be used with negligible drop in performance. Leon and Kumar (2007) presented the details of a mathematical model using heat transfer expressions for the collector components and empirical relationships for estimating the various heat transfer coeﬃcients. A parametric study was carried out for a wide range of designs and operating conditions. Their study concluded that solar
2804
M. Badache et al. / Solar Energy 86 (2012) 2802–2810
absorptivity, collector pitch, and air ﬂow rate had the...

...the self-concepts of students. In this case, the independent variable, counselling approach has three levels. Necessarily there should be three groups randomly selected from the school population which will be exposed to three different counselling approaches.
The dependent variable, self-concept, may be measured through a standardized self-concept instrument which yields interval scores for the subjects.
In this problem, application of the one-factor ANOVA will test the following hypothesis: There is no significant difference in self-concept among the three groups of students exposed to different counselling approaches.
Step 1 Enter the data in a worksheet table. (See below.)
Step 2 Find the square of each raw score (X2).
Step 3 Compute the sum of N for each group, the total N, the sums of the raw scores and the sums of the squared scores.
N1 = 6; N2 = 6; N3 = 6; Nt = 18
ƩX1 = 366; ƩX2 = 492: ƩX3 = 510: ƩXt = 1368
ƩX21 = 23, 866; ƩX22 = 40, 798; ƩX23 = 43, 652; ƩX2t = 108, 316
WORKSHEET TABLE for the One-WayANOVA
Counselling Conditions
Group (1) Peer (2) Individual (3)
X1 X21 X2 X22 X3 X23
78 6084 77 5929 78 6084
46 2116 83 6889 91 8281
41 1681 97 9409 97 9409
50 2500 69 4761 82 6724
69 4761 79 6241 85 7225
82 6724 87 7569 77 5929
SUMS 366 23866 492 40798 510 43652 ƩXt=1368
Means 61 82 85 ƩX2t = 108, 316
N 6 6 6 Nt = 18
Step 4 Compute Sums of Squares.
a. SSt (SS for total variability)...

...Analysis of Variance
(ANOVA)
Dr. H. Johnson
ANOVA
• Analysis of variance (ANOVA) is a powerful hypothesis
testing procedure that extends the capability of t-tests
beyond just two samples.
• Many types of ANOVAs, today we will learn about a oneway independent-measures ANOVA
• Later we’ll learn one-way repeated-measures ANOVA .
• We’ll also learntwo-factor ANOVA after that.
• These ANOVAs are by no means all of them! There are a
LOT more types!
One-WayANOVA
• The independent measures ANOVA is used in the same types of
situations that the independent measures t-test had been used,
except that the ANOVA allows for the comparison of more than
just two groups.
• Before the advent of the computer, if someone had three
groups in an experiment, they would often use a series of t-tests
to compare all possible combinations of means.
• If you had three samples to compare then, using t-tests, we
would have to do M1 vs. M2, M1 vs. M3, and M2 vs. M3
• Each time we do a t-test, the type I error rate is equal to a.
• The experiment-wise error rate (a) is held at .05 in an ANOVA.
One-WayANOVA
Preliminary Example
Pretend you wanted to know the effects of different temperatures on
the ability...

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