In this discussion, you will evaluate a research question and determine how that question might best be analyzed. To do this, you will need to identify the appropriate application of course specified statistical tests, examine assumptions and limitations of course specified statistical tests, and communicate in writing critiques of statistical tests.
A researcher wishes to study the effect of a new drug on blood pressure. Consider and discuss the following questions as you respond:
- Would you recommend using a z-test, a t-test, or an ANOVA for the analysis? Explain your answer.
- What would your choice of test depend on? For the test you select, explain your design and your comparison groups.
- Would the hypothesis be directional or non-directional?
- Would the test be one-tailed or two-tailed?
- What would be the null and what would be the alternative hypothesis?The purpose of a One-way ANOVA:
- When a study looks at three or more groups of scores, a one-way ANOVA permits the researcher to use the data for the purpose of making a single inferential statement concerning the means of the study’s populations.
- No matter how many samples involved there is only one inferential statement made from the set of samples to the set of populations.
- The inferential question is “Are the means of the various populations equal to one another?”
- The focus of the inference is on the population means, even though each sample is described in terms of mean, SD and n.
VassarStats: Website for Statistical Computation (Links to an external site.). (http://vassarstats.net)
- This is a website includes tools to calculate many of the statistical tests we cover in this course including t-tests, ANOVA, correlation, and regression. Each calculator includes a tutorial and/or walkthrough.
Web Center for Social Research Methods (Links to an external site.). (http://socialresearchmethods.net)
- This website includes links to numerous tools and tutorials relating to statistical concepts, calculations, and scale development.
The distinction between One-way ANOVA and other kinds of ANOVAs
There are many synonyms for “one-way ANOVA” including:
- one-way ANOVA, one-factor ANOVA, F-test, or a simple ANOVA
The number of independent variables (i.e., factors) and dependent variables
- a one-way ANOVA has one independent variable (factor)
- it focuses on one dependent variable
- and, it involves samples that are independent
When we say that there is just one independent variable, this means that the comparison groups differ from one another, prior to the collection and analysis of any data, in one manner that is important to the researcher.
- The comparison groups can differ in terms of a qualitative variable (e.g., religious preference)
- Or, in terms of a quantitative variable (e.g., time spent studying)
- But there can be only one characteristic that defines how the comparison groups relate
- Factor and independent variable mean the same thing in ANOVA
- The dependent variable corresponds to the measured characteristic of people, animals, or things from whom or which data are gathered.
The “between” vs. “within” nature of the independent variable
- Between subjects means comparisons are made with data that come from independent samples.
- When data come from correlated samples, the independent variables are considered to be within subjects in nature
The One-way ANOVA’s null and alternative hypothesis
The null hypothesis of a one-way ANOVA is always set up to say that the mean score on the dependent variable is the same in each of the populations associated with the study.Two ways to express the null hypothesis
- The typical way is: H0: μ1 = μ2 = μ3 = μ4
- σ2μ = 0 puts the pinpoint number into view—there is no variability among the means
Presentation of results
The outcome of a one-way ANOVA is presented in one of two ways: a summary table or a sentence or two in the text.
Results of a single one-way ANOVA- Can be displayed in a typical ANOVA summary table.
- By using df values—you can use the first and third df to help you understand the structure of a completed study
- by adding 1 to the between groups df, you can determine, or verify, the number of groups in the study
- by adding 1 to the total df you can figure out the number of participants in the study
- textual presentation of results
- e.g., F(3, 56) = 3.23
- the numbers in parentheses are the df from the between groups and within groups rows of the summary table
- add 1 to the first number to get the number of groups in the study
- to figure the total number of participants you add the two numbers together and add 1
Assumptions of a One-way ANOVA
The four main assumptions (Huck, 2008):- normality assumption
- homogeneity of variance assumption
- randomness assumption
- independence assumption
Checking on the normality and equal variance assumptions
- many researchers who use One-way ANOVA seem to pay little attention to the assumptions
- some researchers use preliminary tests to address the assumptions and transform their data
- when testing the homogeneity of variance assumption, researchers want to retain the null hypothesis of equal population variances so that they can move forward and compare their sample means with a one-way ANOVA
Three options when the assumptions seem untenable
Sometimes, preliminary checks on normality and the equal variance assumption suggest that the populations are not normal and/or have unequal variances. When this happens, researchers have three options (Huck, 2008):- Identify and eliminate outliers, presuming such scores are the problem
- Transform their sample data in an effort to reduce non-normality and/or stabilize the variances
- Or, switch from a one-way ANOVA F-test to some other test that does not have such rigorous assumptions
Independence and the “unit of analysis”
- The independence assumption is the one most neglected
- In essence, this assumption says that a particular person’s score should not be influenced by the measurement of any other people or by what happens to others in the execution phase of the study.
- This assumption would be violated if different groups of students (perhaps different intact classrooms) are taught differently, with each student’s exam score being used in the analysis
- In studies where groups are a necessary feature of the investigation, the recommended way to adhere to the independence assumption is to have the unit of analysis (i.e., the scores are analyzed) be each group’s mean rather than the scores from the individuals in the group.
Do researchers typically worry about practical significance?
- Researchers typically do not take time to address the issue of statistical versus practical significance.
- Many researchers use the simplest version of hypothesis testing to test their one-way ANOVA.
- They collect the amount of data that time, money, or energy will allow, and then they anxiously await the outcome of their analysis.
- If their F-ratios turn out to be significant, these researchers quickly summarize their studies, with emphasis put on the fact that “significant findings” have been obtained (Huck, 2008).
Degrees of FreedomThe team choice analogy
A simple (though not completely accurate) way of thinking about degrees of freedom is to imagine you are picking people to play in a team. You have eleven positions to fill and eleven people to put into those positions.
How many decisions do you have? In fact you have ten, because when you come to the eleventh person, there is only one person and one position, so you have no choice.Thus you have ten ‘degrees of freedom’.Likewise, when you have a sample, the degrees of freedom to allocate people in the sample to tests are one less than the sample size. So if there are N people in a sample, the degrees of freedom are N-1.
Where there are multiple samples, then the degrees of freedom for each are
N1-1,N2-1, etc.When the samples are combined, the total degrees of freedom are
(N1 + N2 + …) – Y, where Y is the number of samples.Thus combining two groups gives DF = N1 + N2 – 2.
If you have two teams and two team captains each of them would have the same situation when then get to the last person—so, degrees of freedom are
N1-1, N2-1 = N-2Reporting Degrees of Freedom
t-Tests are reported with degrees of freedom in parentheses. Following that, report the t statistic (rounded to two decimal places) and the significance level.Example:
There was a significant effect for gender, t(67) = 5.43, p < .001, with women receiving higher scores than menThe ANOVAs (both one-way and two-way) are reported like the t test, but there are two degrees-of-freedom numbers to report.
- First report the between-groups degrees of freedom, and then report the within-groups degrees of freedom (separated by a comma).
- After that, report the F statistic (rounded off to two decimal places to adhere to APA style) and the significance level.
- Example: There was a significant main effect for treatment, F(1, 145) = 5.43, p = .02, and a significant interaction, F(2, 145) = 3.24, p = .04.
- Example: A paired-samples t-test indicated that scores were significantly higher for the pathogen subscale (M = 26.4, SD = 7.41) than for the sexual subscale (M = 18.0, SD = 9.49), t(721) = 23.3, p < .001, d = 0.87 (Huck, 2008).
- Because it was a paired-samples t-test you can tell that N = 722 from the degrees of freedom that is in the parentheses after t (since N-1 all you do as add the 1 back to get 722)
- Example: An independent-samples t-test indicated that scores were significantly higher for women (M = 27.0, SD = 7.21) than for men (M = 24.2, SD = 7.69), t(734) = 4.30, p < .001, d = 0.35 (Huck, 2008).
- Because the test was an independent-samples t-test you can tell that N = 736 from the degrees of freedom that is in the parentheses after t (since N-2 all you do as add the 1 back to get 736)
Yes, there are a lot of numbers here, but do not get overwhelmed. Just take it step by step.
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