Overview
t-Tests,
ANOVAs,
of
&
ANCOVAs
t-Test
The most basic statistical test that measures group differences
is the t test, which analyzes significant differences between two
group means. Consequently, a t test is appropriate when the IV is
defined as having two categories and the DV is quantitative, (e.g.,
What is the difference between males and females [IV] with respect to
SAT scores [DV]? Is this difference statistically significant?). There
are two variations of t-tests: an independent-samples t-test and a
repeated-measures t-test.
When to use a t test?
1 IV (2 categories)
Group differences
1 DV (quantitative)
One-Way Analysis of Variance
One-way analysis of variance (ANOVA) tests the significance of
group differences between two or more means as it analyzes variation
between and within each group. ANOVA is appropriate when the IV is
defined as having three or more categories and the DV is quantitative,
(e.g., Do preschoolers of low, middle, and high socio-economic status
[IV] have different literacy test scores [DV]? If so, which groups are
significantly different?). Since ANOVA only determines the
significance of group differences and does not identify which groups
are significantly different, post hoc tests are usually conducted in
conjunction with ANOVA.
When to use a one-way ANOVA?
1 IV (2+ categories)
Group differences
1 DV (quantitative)
One-Way Analysis of Covariance
One-way analysis of covariance (ANCOVA) is similar to ANOVA in
that two or more groups are being compared on the mean of some DV, but
ANCOVA additionally controls for a variable (covariate) that may
influence the DV (e.g., Do preschoolers of low, middle, and high
socio-economic status [IV] have different literacy test scores [DV]
after adjusting for family type [covariate]?) Many times the
covariate may be pretreatment differences, in which groups are equated
in terms of the covariate(s). In general, ANCOVA is appropriate when
the IV is defined as having two or more categories, the DV is
quantitative, and the effects of one or more covariates needs to be
removed.
When to use a one-way ANCOVA?
1 IV (2+ categories)
Group differences
1 DV (quantitative)
1+ covariate
Source:
Mertler, C. A., & Vannatta, R. A. (2005). Advanced and
multivariate statistical methods: Practical application and
interpretation (3rd ed.). Los Angeles, CA: Pyrczak.
t-Test
I.
Research Questions…



II.

Generically-Stated Research Questions:

Independent-samples: What is the difference between Group 1
and Group 2 on Variable A? Is this difference statistically
significant?

Repeated-measures: What is the difference between Measure 1
(i.e., Var. 1) and Measure 2 (i.e., Var. 2) for this sample?
Is this difference statistically significant?
Examples of Appropriately-Stated Research Questions:

What is the difference between males and females with respect
to SAT scores? Is this difference statistically significant?

What is the difference between pretest and posttest scores for
high school sophomores following an OGT prep course? Is this
difference statistically significant? [ALT: How effective is
an OGT prep course for high school sophomores?]
Examples of Inappropriately-Stated Research Questions:

What is the relationship between gender and SAT scores?

Is there a difference between pretest and posttest scores for
high school sophomores following an OGT prep course?
Sampling & Data…
Samples should always be selected randomly (probability samples)


Data must be quantitative

III.
Allows for generalization of results to larger population,
especially using any inferential statistical test
Scale of measurement (i.e., nominal, ordinal, interval, ratio)
– grouping variable is nominal (categorical); actual measures
must be interval or ratio
Data Analysis & Interpretation…

Analysis involves the calculation and interpretation of a tstatistic (& its associated statistical significance)

Sample output from SPSS…
Descriptive
statistics
for the two
groups
Tests Ho that
the
variances of
the 2
populations
are equal
t-statistic &
significance
(p-value);
if < .05, then
difference
between 2 groups
is significant
One-Way Analysis of
I.
Research Questions…

Generically-Stated Research Question:





Do preschoolers of low, middle, and high socio-economic status
have different literacy test scores? If so, which groups are
significantly different?

How do teachers’ assessment practices differ based on school
district rating? Which of these differences, if any, are
statistically significant?
Examples of Inappropriately-Stated Research Questions:
Samples should always be selected randomly (probability samples)
Allows for generalization of results to larger population,
especially using any inferential statistical test
Data must be quantitative

III.
Similar to those for t-test…
Sampling & Data…


What is the difference between Groups 1, 2, & 3 on Variable A?
Which, if any, groups are significantly different?
Examples of Appropriately-Stated Research Questions:

II.
Variance
Scale of measurement (i.e., nominal, ordinal, interval, ratio)
– grouping variable is nominal (categorical); actual measures
must be interval or ratio
Data Analysis & Interpretation…

Analysis involves the calculation and interpretation of an Fstatistic or F-ratio (& its associated significance)

Sample output from SPSS…
Descriptive
statistics
for the five
groups
Value of
F-ratio
P-value; p <
.05, so there
are
significant
differences…bu
t where???
Also, note
about the
value of .000!
These are
pairwise
comparisons;
every pair
appears twice in
the table
Any value < .05
indicates a
significant
difference between
these 2 particular
groups
One-Way Analysis of
I.
Research Questions…

Generically-Stated Research Question:





Do preschoolers of low, middle, and high socio-economic status
have different literacy test scores after adjusting for family
type?

How do teachers’ assessment practices differ based on school
district rating, controlling for years of teaching experience?
Which of these differences, if any, are statistically
significant?
Examples of Inappropriately-Stated Research Questions:
Samples should always be selected randomly (probability samples)

Allows for generalization of results to larger population,
especially using any inferential statistical test
Data must be quantitative

III.
Similar to those for t-test…
Sampling & Data…


What are the differences between Groups 1, 2, & 3 on Variable
A, after controlling for the effects of Variable B? Which, if
any, groups are significantly different?
Examples of Appropriately-Stated Research Questions:

II.
Covariance
Scale of measurement (i.e., nominal, ordinal, interval, ratio)
– grouping variable is nominal (categorical); actual measures
must be interval or ratio (including covariate)
Data Analysis & Interpretation…
Analysis involves the calculation and interpretation of an Fstatistic or F-ratio (& its associated significance)

Sample output from SPSS…
“District
Rating” is
the DV
“Years of
Experience
” is the
covariate
Significance
of
covariate &
of group
differences
(after
controlling
for the
covariate)