# Inferential Statistics

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Inferential Statistics

Techniques that allow us to study samples and then make generalizations about the population. 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

Uses for Inferential Statistics

Statistics used in descriptive research when comparisons are made between different groups

Uses for Inferential Statistics

These statistics enable the researcher to evaluate the effects of an independent variable on a dependent variable

Sampling Error

Sample does not represent the population

Hypothesis Testing Procedures

State the hypothesis (H0)

Select the probability level (alpha)

Determine the value needed for significance

Calculate the test statistic

Accept or reject H0

Select the probability level (alpha)

Determine the value needed for significance

Calculate the test statistic

Accept or reject H0

Statistical Significance

statistical assessment of whether observations reflect a pattern rather than just chance

Parameter Statistics

Techniques which require basic assumptions about the data, for example:

normality of distribution

homogeneity of variance

requirement of interval or ratio data

normality of distribution

homogeneity of variance

requirement of interval or ratio data

t-tests

requires interval or ratio level scores

used to compare two mean scores

easy to compute

pretty good small sample statistic

used to compare two mean scores

easy to compute

pretty good small sample statistic

Independent Groups t-test

compares mean scores on two independent samples

Dependent Groups (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

most common situation is for comparison of pretest with posttest scores from the same sample

Hypothesis Testing Error (Type I)

made when the researcher rejects the null hypothesis when in fact the null hypothesis is true

probability of committing Type I error is equal to the significance (alpha) level set by the researcher

thus, the smaller the alpha level the lower the chance of committing a Type I error

probability of committing Type I error is equal to the significance (alpha) level set by the researcher

thus, the smaller the alpha level the lower the chance of committing a Type I error

Hypothesis Testing Error (Type II)

occurs when the researcher accepts the null hypothesis, when in fact it should have been rejected

probability is equal to beta (B) which is influenced by several factors

inversely related to alpha level

increasing sample size will reduce B

probability is equal to beta (B) which is influenced by several factors

inversely related to alpha level

increasing sample size will reduce B

Statistical Power

the probability of rejecting a false null hypothesis

Power = 1 – beta

Decreasing probability of making a Type II error increases statistical power

Power = 1 – beta

Decreasing probability of making a Type II error increases statistical power

ANOVA

Analysis of Variance; Extension of t-test

-requires interval or ratio level scores

used for comparing 2 or more mean scores

maintains designated alpha level as compared to experimentwise inflation of alpha level with multiple t-tests

may also test more than 1 independent variable as well as interaction effect

-requires interval or ratio level scores

used for comparing 2 or more mean scores

maintains designated alpha level as compared to experimentwise inflation of alpha level with multiple t-tests

may also test more than 1 independent variable as well as interaction effect

One-way ANOVA

Extension of independent groups t-test, but may be used for evaluating differences among 2 or more groups

Repeated Measures ANOVA

Extension of dependent groups t-test, where each subject is measured on 2 or more occasions

a.k.a “within subjects design”

a.k.a “within subjects design”

Random Blocks ANOVA

This is 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 and then randomly assigned to treatment groups

Participants similar in terms of a variable are placed together in a block and then randomly assigned to treatment groups

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)

Two-way ANOVA (e.g., 3 X 2 ANOVA)

Three-way ANOVA (e.g., 3 X 3 X 2 ANOVA)

Assumptions of Statistical Tests

Interval or ratio level scores

Random sampling of participants

Scores are normally distributed

N = 30 considered minimum by some

Homogeneity of variance

Groups are independent of each other

Others

Random sampling of participants

Scores are normally distributed

N = 30 considered minimum by some

Homogeneity of variance

Groups are independent of each other

Others

Two-Group Comparison Tests

The various ANOVA tests are often referred to as “omnibus” tests because they are used to determine if the means are different but they do not specify the location of the difference

if the null hypothesis is rejected, meaning that there is a difference among the mean scores, then the researcher needs to perform additional tests in order to determine which means (groups) are actually different

if the null hypothesis is rejected, meaning that there is a difference among the mean scores, then the researcher needs to perform additional tests in order to determine which means (groups) are actually different

Multiple Comparison (post hoc) tests

Duncan

Tukey

Bonferroni

Scheffe

Tukey

Bonferroni

Scheffe

Analysis of Covariance

ANOVA design which statistically adjusts the difference among group means to allow for the fact that the groups differ on some other variable

frequently used to adjust for inequality of groups at the start of a research study

frequently used to adjust for inequality of groups at the start of a research study

Nonparametric Statistics

Considered assumption free statistics

Appropriate for nominal and ordinal data or in situations where very small sample sizes (n < 10) would probably not yield a normal distribution of scores

Less statistical power than parametric statistics

Appropriate for nominal and ordinal data or in situations where very small sample sizes (n < 10) would probably not yield a normal distribution of scores

Less statistical power than parametric statistics

Chi Square

A nonparametric test used with nominally scaled data which are common with survey research

The statistic is used when the researcher is interested in the number of responses, objects, or people that fall in two or more categories

The statistic is used when the researcher is interested in the number of responses, objects, or people that fall in two or more categories

Single Sample Chi-Square (goodness of fit)

Used to test the hypothesis that the collected data (observed scores) fits an expected distribution

Independent Groups Chi-Square (contingency table)

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

In this test we are comparing two or more patterns of frequencies to see if they are independent from each other

Univariate Statistic

used in situations where each participant contributed one score to the data analysis, or in the case of a repeated measures design, one score per cell

Multivariate Statistic

used in situations where each participant contributes multiple scores

Multivariate Tests (MANOVA)

Canonical correlation

Discriminant analysis

Factor analysis

Discriminant analysis

Factor analysis

Multiple Analysis of Variance (MANOVA)

Analogous to ANOVA except that there are multiple dependent variables

Represents a type of multivariate test

Represents a type of multivariate test

Simple Prediction

Predicting an unknown score (Y) based on a single predictor variable (X)

Y’ = bX + c

Y’ = bX + c

Multiple Prediction

Involves more than one predictor variable

Y’ = b1X1 + b2X2 + c

Y’ = b1X1 + b2X2 + c

Multiple Regression/Prediction (Multiple Correlation)

Determines the relationship between one dependent variable and 2 or more predictor variables

Used to predict performance on one variable from another

Y’ = b1X1 + b2X2 + c

Standard error of prediction is an index of accuracy of the prediction

Used to predict performance on one variable from another

Y’ = b1X1 + b2X2 + c

Standard error of prediction is an index of accuracy of the prediction

Statistical Power

alpha = probability of a Type I error

rejecting a true null hypothesis

this is your significance level

beta = probability of a Type II error

failing to reject a false null hypothesis

Statistical power = 1 – beta

rejecting a true null hypothesis

this is your significance level

beta = probability of a Type II error

failing to reject a false null hypothesis

Statistical power = 1 – beta

Factors Affecting Power

Alpha level

Sample size

Effect size

One-tailed or two-tailed test

Sample size

Effect size

One-tailed or two-tailed test

Alpha Level

Reducing the alpha level (moving from .05 to .01) will reduce the power of a statistical test. This makes it harder to reject the null hypothesis

Sample Size

In general, the larger the sample size the greater the power. This is because the standard error of the mean decreases as the sample size increases

One-tailed vs. two-tailed tests

It is easier to reject the null hypothesis using a one-tailed test than a two-tailed test because the critical region is larger

Effect Size

This is an indication of the size of the treatment effect, its meaningfulness

With a large effect size, it will be easy to detect differences and statistical power will be high

But, if the treatment effect is small, it will be difficult to detect differences and power will be low

With a large effect size, it will be easy to detect differences and statistical power will be high

But, if the treatment effect is small, it will be difficult to detect differences and power will be low

Effect Size

ES=(M1-M2)/SD

Small ES

0.2

Moderate ES

0.5

Large ES

0.8

Qualitative methods

focus on understanding and explaining meaning of a social phenomena

Qualitative

Subjective

Non-numerical

Nonstatistical analysis

Small Ns

Open ended data collection

Narrative for results

Non-numerical

Nonstatistical analysis

Small Ns

Open ended data collection

Narrative for results

8 Characteristics of qualitative research

-Takes place in the natural setting: travel to sites

-Researcher is the primary method of data collection

-Emergent rather than tightly prefigured

-Based upon interpretation

Hermeneutics: deciphering meaning

-Views social phenomena holistically

-Qualitative researchers reflect and are explicit regarding personal assumptions and values

-Uses both deductive and inductive logic

Inductive: going from specific to large

Deductive: Going from broad to specific

-Can use multiple methods

-Researcher is the primary method of data collection

-Emergent rather than tightly prefigured

-Based upon interpretation

Hermeneutics: deciphering meaning

-Views social phenomena holistically

-Qualitative researchers reflect and are explicit regarding personal assumptions and values

-Uses both deductive and inductive logic

Inductive: going from specific to large

Deductive: Going from broad to specific

-Can use multiple methods

Qualitative Methods Types

Life histories

Grounded Theory Study

Case Study

Phenomenology Study

Ethnography Study

Basic/Generic

Grounded Theory Study

Case Study

Phenomenology Study

Ethnography Study

Basic/Generic

Life Histories

Story of a single individual or groups of single individuals

Grounded Theory Study

Discover or invent theory grounded in real-world experiences

Middle-range theories: situation related

Middle-range theories: situation related

Case Study

Exploration of a bounded system (e.g., school)

In-depth data collection involving multiple sources of information

In-depth data collection involving multiple sources of information

Phenomenology Study

Describes the meaning of a lived experience for several individuals about a phenomenon

Explores the structures of human consciousness

Explores the structures of human consciousness

Ethnography Study

Interpretation of a cultural or social group

Natural setting

Natural setting

Basic/Generic

Studies that illustrate characteristics of qualitative research

Complete Participation

Researcher conceals role

Observer as participant

role of researcher is known

participant as observer

observational role is secondary to participant role

complete observer

researcher observes without participating

constant comparison

technique for analyzing qualitative data