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Chapter 15

- Inferential Data Analysis

Inferential Statistics

- Inferential statistics are a very crucial part

of scientific research in that these techniques

are used to test hypotheses.

Uses for Inferential Statistics

- Statistics for determining differences between

experimental and control groups in experimental

research - Statistics are also used in descriptive research

when comparisons are made between different

groups

- These statistics enable the researcher to

evaluate the effects of an independent variable

on a dependent variable.

Sampling Error

- Remember when we talked about the sampling error?
- Parameters characteristics of a population
- Statistics characteristics of a sample

- Differences between a sample statistic and a

population parameter because the sample is not

perfectly representative of the population. - So, maybe the differences among the sample means

may be a real difference, or it may be due to

sampling error.

- Researcher needs a standard procedure to follow

in making such a decision.

Hypothesis-testing procedure

Hypothesis testing

- Hypothesis testing or significance testing is

used to determine whether what you observed in

the sample provides enough evidence to believe

that there is a difference in the population. - In other words
the sample difference is said to

be statistically different or not statistically

different.

Hypothesis Testing

- The Research Hypothesis is transformed into a

Statistical or Null Hypothesis. - This is done so that statistical tests can be

employed that will determine whether the findings

are statistically significant or can be

attributed to chance. - The results of the statistical test will enable

the researcher to accept or reject the null

hypothesis.

More Hypothesis Testing

- The purpose of the statistical test is to

evaluate the null hypothesis at a specified level

of probability - For instance, testing the difference in the mean

values between 2 groups at the .05 level means

- Do the values of the dependent variable differ

significantly (plt.05) so that these differences

would not be attributable to chance occurrence

more than 5 times in 100?

Level of Significance

- Rejecting the null hypothesis at the .05 alpha

(?) level suggests a 95 probability that the

differences between the two variables is real and

not the result of chance.

- Type I Error rejecting a null hypothesis when it

is really true. - Probability of making a type I error is equal to

? - Type II Error acceptance or not rejecting a null

hypothesis when it is false. - Probability of making a type II error is equal

to ß.

Hypothesis Testing Procedures

- State the hypothesis (H0), and then you select

the probability level (alpha). - The researcher sets a significance level to

indicate the maximum risks she is willing to take

of making an error when concluding that there is

a difference attributable to the research

situation and not to chance. - Commonly used ? 0.05 or 0.01.

- Next, you should decide if you are going to use a

one-tailed or a two-tailed test. - In a one-tail test, the 5 area of rejection is

either at the upper end or the lower end.

To reject the null, the tail used for the

rejection region should cover the extreme

The t or z scores that are rejected are ones in

the red region or positive values.

- In a two-tail test, the 5 area of rejection is

split between the upper and lower tails of the

curve null hypothesis is nondirectional.

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- The next thing is that you need to determine if a

parametric test or a nonparametric test should be

used. - Most common mistakes doing a parametric test

when the data are not normally distributed.

Parametric Nonparametric Tests

- Parametric
- Assume the data are normally distributed.
- That the dependent variables are continuous and

measured on an interval or ratio scale. - Parametric tests are powerful tools make maximum

use of the information available in the data

(e.g. mean, standard deviation).

Normal curve

- The normal curve is a statistical model that is

used to visualize data, interpret distributions

of scores, and make predictions and probability

statements. - Mean, median, and mode are identical, and makes

up the vertical midpoint. - 95 of the area is between 2 SD.

Normal Curve

- Nonparametric
- Make no assumptions about the population under

investigation. - Can be used with nominal or ordinal data.
- Can be used for very small samples, or large

samples which do not fit parametric test

assumptions.

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- Major drawback with these tests is that they are

less powerful than their parametric analogs. - Increased chance of a Type II error less

sensitive to small differences and less able to

detect that such differences might be

statistically significant.

- Check
- Data is normally distributed?
- Population variances of the groups are

approximately equal? - That the dependent variables are continuous and

measured on an absolute, interval, or ratio

scale? - Large or small sample size?

- If youre not sure
there are a number of

statistical tests to see if your data is

parametric or nonparametric. These tests are

called tests for normality or goodness of fit

test. - Eg Kuipers goodness of fit
- Watsons goodness of fit Liliefors test for

normality - Kolmogorov-Smirnov goodness of fit

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To reiterate yesterdays class

- Hypothesis-testing procedure
- State the hypothesis
- Select the probability level (?)
- Decide if the data are normal or not normal.
- Consult the statistical table.

Parametric Tests

t-tests

- Characteristics of t-tests
- requires interval or ratio level scores
- used to compare two mean scores
- easy to compute
- pretty good small sample statistic

Types of t-test

- One-Group t-test
- t-test between a constant and a sample mean
- Independent Groups t-test
- compares mean scores on two independent samples

Total and saturated fat intake by year of study.

Total and saturated fat intake as affected by

major and year of study.

- Dependent Groups or paired (Correlated) t-test
- compares two mean scores from a repeated measures

or matched pairs design - most common situation is for comparison of

pretest with posttest scores from the same sample

Analysis of Variance (ANOVA)

- Analysis of Variance compares two or more means

to determine if there are statistically

significant differences among them. - Compares the amount of variation between the

groups with the variation within the groups. - If ANOVA determines that differences exist, data

are then further tested with post hoc test

- Examples of post hoc tests
- Duncans multiple range test
- Student-Newman-Keuls test (SNK)
- Tukeys Honestly Significant Difference (HSD)
- Scheffes test
- These tests determine how the individual means

differ.

One-way ANOVA

- Extension of independent groups t-test, but may

be used for evaluating differences among 2 or

more groups.

Repeated Measures ANOVA

- Each subject is measured on 2 or more occasions
- a.k.a within subjects design.
- Example three Ca dosage, measure 5 times each

Random Blocks ANOVA

- An extension of the matched pairs t-test when

there are three or more groups or the same as the

matched pairs t-test when there are two groups. - Participants similar in terms of a variable are

placed together in a block, then randomly

assigned to treatment groups another source of

variation that you want to eliminate

- Example three doses of Ca given to subjects

from ethnic groups. Assign subjects from each

ethnic group to each treatment.

Factorial ANOVA

- This is an extension of the one-way ANOVA for

testing the effects of 2 or more independent

variables as well as interaction effects. - Two-way ANOVA (e.g., 3 X 2 ANOVA)
- Three-way ANOVA (e.g., 3 X 3 X 2 ANOVA)
- Example 3 doses of Ca (1st variable) and sex of

the subjects (2nd variable).

Multivariate Analysis of Variance (MANOVA)

- MANOVA determines whether differences exist

between two or more treatments considering

several variables at the same time. - Replaces repeating ANOVA on each variable
- Determines whether treatments differ on balance

considering all variables at once (e.g. blood

chemistry).

Nonparametric Tests

The Chi-Square Test

- Used to test the distribution of observations in

a sample among classes - Determines whether a single sample is different

from a hypothetical distribution (e.g., normal

distribution), or whether two or more

distributions differ from each other. - Example incidence of obesity (on a 1-5 scale -

counts) in men and women

Single Sample Chi-Square

- a.k.a one-way chi-square or goodness of fit

chi-square - Used to test the hypothesis that the collected

data (observed scores) fits an expected

distribution

Independent Groups Chi-Square

- a.k.a. two-way chi-square or contingency table

chi-square - Used to test if there is a significant

relationship (association) between two nominally

scaled variables - In this test we are comparing two or more

patterns of frequencies to see if they are

independent from each other

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The Mann-Whitney U-Test

- Nonparametric equivalent of a t-test requires no

assumptions about the nature of the data.

The Wilcoxon Matched-Pair Signed Rank Test

- An nonparametric test equivalent to the t-test

for paired samples - Applied when each sample from one treatment is

matched with a sample from the other treatment

(by subject age, for example)

The Kruskal-Wallis Test

- The nonparametric equivalent of a one-way or

paired Analysis of Variance

Finally

- Conduct the statistical test

- Accept or reject the null hypothesis.
- If the p value for the value of the statistical

test is less than the alpha level, the null

hypothesis is rejected. - If the opposite is true, the null hypothesis is

accepted.

Hypothesis Testing Errors

- Remember, the researcher sets a significance

level to indicate the maximum risks she is

willing to take of making an error when

concluding that there is a difference

attributable to the research situation and not to

chance. - Commonly used ? 0.05 or 0.01.

- Hypothesis testing decisions are made without

direct knowledge of the true circumstance in the

population. As a result, the researchers

decision may or may not be correct.

Possible Decisions in Statistical Significance

Testing