Independent t-test
Features:
• One Independent Variable
• Two Groups, or Levels of the
Independent Variable
• Independent Samples
(Between-Groups): the two
groups are not related in any
way
• Interval/Ratio Measurement of
the Dependent Variable
Basic Concepts:
• The t-test divides the difference
between the two means by the
amount of variation within and
between each of the two
groups
• The t test statistic is normally
distributed
• The tcv is determined based on
the level of significance and the
sample size (Degrees of
Freedom)
• As degrees of freedom
increase, the t-distribution
becomes increasingly normal.
Hypothesis Testing: Independent
Groups t-Ratio
• Computational Formula, Unequal N
X 1 - X 2 - (μ 1 - μ 2 )
t=
√
ΣX 1 2 -
(ΣX 1 ) 2
n1
+ ΣX 2 2 -
(ΣX 2 ) 2
n2
n1 + n2 - 2
• Computational Formula, Equal N
t=
(X 1 - X 2 ) - (μ 1 - μ 2 )
ŝ12
ŝ22
+
n1
n2
√
(
1
1
+
n1
n2
)
Interpreting the t-Ratio
• d2: the effect size statistic for the two-sample t-ratio
|μ1 - μ2|
d2 =
σ
Non-Directional
OR
d2 =
μ1 - μ2
σ
Directional
• Omega Squared (ω2): a statistical index of the degree to
which the IV accounts for the variance in the DV
The Independent t-Ratio and Its
Assumptions
•
Parametric Test: an inferential statistic that makes
specific assumptions about the population(s) from
which the data are drawn
1. The samples are randomly selected; the scores are
normally distributed
2. The samples are independent; the participants have
been randomly assigned to groups
3. The distribution of error is the same across groups;
homogeneity of variance
Computing the Independent t-test Using
SPSS
• Enter data in the data editor or download the file.
• Click analyze  compare means  independent sample
t-test. The independent samples t-test dialog box should
open.
• Click on the independent variable, and click the arrow to
move it to the grouping variable box.
• Highlight the variable (? ?) in the grouping variable box.
Click define groups. Enter the value 1 for group 1 and 2 for
group 2. Click continue.
• Click on the dependent variable, and click the arrow to
place it in the test variable(s) box.
Interpreting the Output
Group Statistics
Overall GPA
anx_cat
Low
High
N
10
10
Mean
3.3800
2.9900
Std. Deviation
.45959
.58205
Std. Error
Mean
.14533
.18406
The group statistics box provides the mean, standard deviation,
and number of participants for each group (level of the IV).
Interpreting the Output
Independent Samples Test
Levene's Test for
Equality of Variances
F
Overall GPA
Equal variances
assumed
Equal variances
not assumed
1.167
Sig.
.294
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
1.663
18
.114
.39000
.23452
1.663
17.081
.115
.39000
.23452
Levene’s test is designed to compare the equality of the error variance of the
dependent variable between groups. We do not want this result to be significant. If
it is significant, we must refer to the bottom t-test value.
t is your test statistic and Sig. is its two-tailed probability. If testing a one-tailed
hypothesis, we need to divide this value in half. df provides the degrees of
freedom for the test, and mean difference provides the average difference in
scores/values between the two groups.
Computing the Independent t-test Using
Excel
• Enter the data for group 1 in column A; enter the data for group 2 in
column B.
• Go to tools  data analysis.
• Highlight t-test: Two samples assuming equal variances. Click OK.
• Click in the variable one range window and highlight the data for
group1. Do the same for group 2.
• Click in the hypothesized mean difference box and enter 0.
• Under output options, click the radio button to the left of output
range. Click in the output range box and highlight an area of your
spreadsheet to the side or below your data (about 15 rows by 3
columns).
• Click OK. Adjust the width of the columns so you can see all of the
information.
t-tests and Variability
• Treatment Effects: conditions created by the IV that may
cause groups to differ from one another (between-groups
variance)
• Random Effects: a variety of factors that influence the
differences among subjects (within-groups variance)
– Measurement Error: a broad class of errors that can influence the
value of an observation; control for as many as possible
– Individual Differences: characteristics of subjects that existed
before the experiment and can influence the results; random
assignment does not always ensure equality of individual
differences
Correlated Groups Design
• Standard Error of the Difference Between Means for the
independent groups design:
• More General Formula for the Standard Error of the
Difference Between Means:
• The r in the equation represents the correlation between
groups. In the independent groups design, the correlation
between groups should be 0.
Correlated Groups Design
• Correlated Groups Designs: designs in which we
do not randomly assign participants and there is an
obvious relation between the two groups
– Repeated Measures Design: the same subject is
tested on two or more occasions (before-after)
– Matched-Group Design: participants are matched on a
variable that is believed to be related to the DV
Dependent (or Paired Samples) t-test
• Has all of the features of the independent t-test
except that the samples are dependent (paired;
correlated).
• t-Ratio for dependent groups:
Computing the Paired Samples t-test Using
SPSS
• Enter data in the data editor or open the file.
• Click analyze  compare means  paired
samples t-test. The paired samples t-test dialog box
should open.
• Hold down the shift key and click on the set of
paired variables (the variables representing the data
for each set of scores). Click the arrow to move
them to the paired variables box.
• Click OK.
Interpreting the Output
Paired Samples Statistics
Pair
1
before
after
Mean
4.8571
8.5000
N
14
14
Std. Deviation
1.70326
1.09193
Std. Error
Mean
.45522
.29183
The paired samples statistics box provides the mean, standard
deviation, and number of participants for each measurement time.
Pa ired Sa mpl es Test
Paired Differenc es
Pair 1
before - aft er
Mean
-3. 64286
St d. Deviat ion
1.59842
St d. Error
Mean
.42720
95% Confidenc e
Int erval of t he
Difference
Lower
Upper
-4. 56576
-2. 71996
t
-8. 527
df
13
Sig. (2-tailed)
.000
The test statistic box provides the mean difference between the two test times, the
t-ratio associated with this difference, its two-tailed probability (Sig.), and the
degrees of freedom associated with the design. As with the independent t-test, if
testing a one-tailed hypothesis, divide the significance level in half.
Entering Between-Groups vs. WithinGroups Data in SPSS
• In a between-groups design (independent t-test, one-way
ANOVA, etc.):
– The independent variable is entered in one column with value
labels for the different levels of the IV
– The values of the dependent variable are entered in one column.
– With one IV and one DV, there will only be two columns of data,
regardless of the number of levels of the IV.
• In a within-groups design (dependent t-test, repeated
measures ANOVA, etc.):
– There are as many columns of data as there are levels of the IV.
– The names of the columns should correspond with the levels of
the IV.
– The score/value for each measurement time is entered in the
appropriate column.
Independent vs. Dependent Groups t-Ratio
• In a dependent groups design, if you ignore the fact
that groups are correlated and use the independent
groups formula, the t-ratio will always be smaller.
• This is because the error term in the denominator
will be larger since the error due to individual
differences has been ignored in the formula and has
not been removed.