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EDUCATIONAL RESEARCH AND STATISTICAL ANALYSIS - Lesson Content 1-5

The purpose of the previous research and descriptive statistic information was to provide a focused foundation for applied practitioner quantitative research. The content posted to the left will provide support for the researcher the process of analysis of your data using specified statistical tools. For this course, the data is analyzed and and displayed through use of the use of the below technology. Bold Blue Text indicate additional information through a live web link.

StatCrunch

An Introduction to StatCrunch 3.0

STATLETS™

An Introduction to STATLETS

Microsoft Excel

An Introduction to Excel as a Statistical Tool

Again, this instructional content is presented to provided focus to the student as you review you readings both in the text and through the online modules.

Click Below to Review Descriptive Research Foundation Concepts

Educational - Practitioner Research
Lesson One: Terminology for statistics in educational research
Lesson Two: Sampling Procedures
Lesson Three: Descriptive Statistics
Lesson Four: Measures of Central Tendency
Lesson Five: The Normal (Bell) Curve

Lesson 6 - Hypothesis Testing

Hypothesis Testing

Definitions:

Null Hypothesis ( H0 )
Statement of zero or no change. If the original claim includes equality (<=, =, or >=), it is the null hypothesis. If the original claim does not include equality (<, not equal, >) then the null hypothesis is the complement of the original claim. The null hypothesis always includes the equal sign. The decision is based on the null hypothesis.

Alternative Hypothesis ( H1 or Ha )
Statement which is true if the null hypothesis is false. The type of test (left, right, or two-tail) is based on the alternative hypothesis.

Type I error
Rejecting the null hypothesis when it is true (saying false when true). Usually the more serious error.

Type II error
Failing to reject the null hypothesis when it is false (saying true when false).

Test statistic
Sample statistic used to decide whether to reject or fail to reject the null hypothesis.

Critical region
Set of all values which would cause us to reject H0

Critical value(s)
The value(s) which separate the critical region from the non-critical region. The critical values are determined independently of the sample statistics.

Significance level
The probability of rejecting the null hypothesis when it is true. alpha = 0.05 and alpha = 0.01 are common. If no level of significance is given, use alpha = 0.05. The level of significance is the complement of the level of confidence in estimation.

Decision
A statement based upon the null hypothesis. It is either "reject the null hypothesis" or "fail to reject the null hypothesis". We will never accept the null hypothesis.

Conclusion
A statement which indicates the level of evidence (sufficient or insufficient), at what level of significance, and whether the original claim is rejected (null) or supported (alternative).

Lesson 7: t-test

Student's t-Tests

http://www.physics.csbsju.edu/stats/t-test.html

We use this test for comparing the means of two treatments, even if they have different numbers of replicates. In simple terms, the t-test compares the actual difference between two means in relation to the variation in the data (expressed as the standard deviation of the difference between the means.

Table of t-statistics

Choosing an analysis to compare two groups

Paired or unpaired test?

When choosing a test, you need to decide whether to use a paired test. Choose a paired test when the two columns of data are matched. Here are some examples:

You measure a variable (perhaps, weight) before an intervention, and then measure it in the same subjects after the intervention.

You recruit subjects as pairs, matched for variables such as age, ethnic group and disease severity. One of the pair gets one treatment, the other gets an alternative treatment.
You run a laboratory experiment several times, each time with a control and treated preparation handled in parallel.

You measure a variable in twins, or child/parent pairs.

More generally, you should select a paired test whenever you expect a value in one group to be closer to a particular value in the other group than to a randomly selected value in the other group.

Lesson 8: z-test

Z-test for samples

Calculation Computation

What is it?

The z-test for one sample uses a similar formula to the z-score for an individual, and it is used to find significant differences between a sample mean and a population mean. It indicates how likely it is that a sample with a certain mean and standard deviation came from the population being studied—that it has a population mean. Essentially, it asks: Given a population with a certain mean, how likely is it I could draw a sample with my given sample mean?

Lesson 9: ANOVA - ANalysis Of VAriance between groups

ANOVA - What is It?

An ANOVA (Analysis of Variance), sometimes called an F test, is closely related to the t test. The major difference is that, where the t test measures the difference between the means of two groups, an ANOVA tests the difference between the means of two or more groups.

A one-way ANOVA, or single factor ANOVA, tests differences between groups that are only classified on one independent variable. You can also use multiple independent variables and test for interactions using factorial ANOVA (see below). The advantage of using ANOVA rather than multiple t-tests is that it reduces the probability of a type-I error. Making multiple comparisons increases the likelihood of finding something by chance—making a type-I error. Let’s use socioeconomic status (SES) as an example. I have 8 levels of SES and I want to see if any of the eight groups are different from each other on their average happiness. In order to compare all of the means to each other, you would have to run 28 t tests. If your alpha is set at .05 for each test, times 28 tests, the new p is 1.4—you are virtually assured of making a type-I error. So, you across all those 28 tests you would find some significant differences between groups, but there are likely due to error. An ANOVA controls the overall error by testing all 8 means against each other at once, so your alpha remains at .05.

One potential drawback to an ANOVA is that you lose specificity: all an F tells you is that there is a significant difference between groups, not which groups are significantly different from each other. To test for this, you use a post-hoc comparison to find out where the differences are – which groups are significantly different from each other and which are not.

Some commonly used post-hoc comparisons are Scheffe’s and Tukey’s.

Lesson 10: Correlation

Correlation, the relationship between two variables, is closely related to prediction.

The greater the association between variables, the more accurately we can predict the outcome of events. There is rarely an exact correlation of observed results with a mathematical function - the points never fit exactly on the line. The question is therefore whether an association between two variables could have occurred by chance.

Correlation Coefficient

Scatter Plot and Line of Best Fit


There are numerous methods for calculating correlation, e.g:
The parametric Pearson, or "r value", correlation
The nonparametric Spearman correlation

Pearson correlation calculations are based on the assumption that both X and Y values are sampled from populations that follow a normal (Gaussian) distribution, at least approximately, although with large samples, this assumption is not too important.

Alternatively, the nonparametric Spearman correlation is based on ranking the two variables, and so makes no assumption about the distribution of the values.

A correlation analysis is performed in the same as any other statistical test of significance:

Formulate the null hypothesis. A simpler hypothesis has priority over a more complex theory, so the null hypothesis (H0) is therefore that "There is no correlation between the datasets". You also need to set the significance level (a) before performing the test (e.g. 0.05).



Warning:Correlation tests are in some ways the most misused of all statistical procedures!They are able to show whether two variables could be connected. However, they are not able to show that the variables are not connected! If one variable depends on another, i.e. there is a causal relationship, then it is always possible to find some kind of correlation between the two variables. However, if both variables depend on a third, they can show a correlation without any causal dependency between them. Take care!

Example:
There is a direct correlation between the number of mobile phone masts and the decline in the numbers of house sparrows, Passer domesticus. But do mobile phone masts harm sparrows, or are both effects caused by something else? Or are they both completely independent observations which just happen to correlate? We don't know because correlation tests do not reveal this information - further investigation is necessary.



Critical Values of the Correlation Coefficient