# Inferential Statistics

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
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
t-tests
requires interval or ratio level scores
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
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
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
Statistical Power
the probability of rejecting a false null hypothesis
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
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”
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
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)
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
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
Multiple Comparison (post hoc) tests
Duncan
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
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
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
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
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
Multiple Analysis of Variance (MANOVA)
Analogous to ANOVA except that there are multiple dependent variables
Represents a type of multivariate test
Simple Prediction
Predicting an unknown score (Y) based on a single predictor variable (X)
Y’ = bX + c
Multiple Prediction
Involves more than one predictor variable
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
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
Factors Affecting Power
Alpha level
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
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
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
Qualitative Methods Types
Life histories
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
Case Study
Exploration of a bounded system (e.g., school)
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
Ethnography Study
Interpretation of a cultural or social group
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