Correlations and t scores (2)

Correlations and t Scores
Dr. Pedro L. Martinez
Introduction to Educational
Research and Assessment

Populations and Samples
 Population - all the individuals of interest in a particular study
(characteristic that describes a population is called a
parameter)
 Sample - set of individuals selected from a population, usually
intended to represent the population (characteristic that
describes a sample is called a statistic)
 Understanding the relationship between the population and a
sample from that population is always important in
understanding the use of inferential statistics and why
formulas may differ for each of them why don’t we just study
the population?

Populations and Samples
 Population - all the individuals of interest in a particular
study (characteristic that describes a population is called a
parameter)
 Sample - set of individuals selected from a population,
usually intended to represent the population characteristic
that describes a sample is called a statistic)
 Understanding the relationship between the population
and a sample from that population.
 If we are mainly interested in the population, why don’t we
just study the population?

Variables and Data
 Variable - characteristic or condition
that changes or has different values for
different individuals.
 Data - measurements or observations
(plural)
 Data set - collection of measurements or
observation
 Datum - single measurement (also called
score or raw score)

Inferential Statistics
 Inferential Statistics - techniques that allow us
to study samples and then make
generalizations about the populations from
which they were selected.
 z scores helped us to make comparisons
between different distributions by
standardization
 Sampling error - discrepancy that exists
between a sample statistic and the
corresponding population parameter

Research Methods
 Correlational Method - two different variables
are observed to determine whether there is a
relationship between them
 Experimental Method - one variable is
manipulated while another variable is
observed and measured. Used to establish
cause-and-effect relationship between
variables

Correlational Designs
 Represents the strength of the relationship between two
variables
 e.g., # of hours of studying and test grades
 e.g., motivation and persistence
 e.g., ability to delay gratification as a child and success in college
 Scatter plot
 Correlation coefficient (“r”) ranges from -1 to +1
 e.g., r = +.34
 e.g., r = -.52

Interpreting Correlations
 Positive correlation
 increase in studying associated with increase in
tests grade scores
 Negative correlation
 increase in studying associated with decrease in
test grade scores
 No correlation (0 correlation)
 Variables are not related
 Correlation may exist but it might be associated
with another variable (e.g. persistence and
number of hour studying a subject)

Correlations: Positive,
Negative, and None

Correlation ≠Correlation ≠
CausationCausation

Why can’t we infer causality?
 Reverse-Causality Problem
X Y→ or Y X ????←
Is there a relationship
between exposure to
violent films and
aggression?

Why can’t we infer
causality?
 Reverse-Causality Problem
X Y→ or Y X ????←
 Third-variable problem
A X→ and A Y→
e.g., ice cream sales and violence (r =
+.29)
VERY IMPORTANT FOR

Explaining Correlations: Three
Possibilities

Not Causality
It is essential that you remember this point: A
nonexperimental research study could be
seriously flawed if you are interested in
concluding that an observed relationship is a
causal relationship.
 That's because "observing a relationship
between two variables is not sufficient grounds
for concluding that the relationship is a causal
relationship." (Remember this important point!)

The Three Necessary Conditions
for Cause-and-Effect Relationships
 It is essential that your remember that
researchers must establish three
conditions if they are to make a defensible
conclusion that changes in variable A
cause changes in variable B. Here are the
conditions (which have been agreed upon
by reserachers in a summary table:

Three Necessary Conditions for
Causation
Condition #1 Condition #2 Condition #3
Variable A and Variable B
must be related
Proper time sequence
must be established
(temporal antecedent
condition)
The relationship between
Variables A & B must not
be through a confounding
extraneous variable (the
third variable and the lack
of an alternative
explanation condition )

Applying the Three Necessary
Conditions for Causation in
Nonexperimental Research
 Non-experimental research is much weaker than
strong and quasi experimental research for
making justified judgments about cause and
effect.
 It is, however, quite easy to establish condition
1 in non-experimental research—just see if the
variables are related For example, Are the
variables correlated? or Is there a difference
between the means?
 It is much more difficult to establish conditions 2
and 3 (especially 3).

Designs Used to Avoid Pitfalls
 When attempting to establish condition 3,
researchers use logic and theory (e.g., make
a list of extraneous variables that you want to
measure in your research study), control
techniques (such as statistical control and
matching), and design approaches (such as
using a longitudinal design rather than a
cross-sectional design).

Real Life Example
• Here is one more example of controlling for a variable: There is
a relationship between gender and income in the United States.
In particular, men earn more money than women. Perhaps this
relationship would disappear if we controlled for the amount of
education people had. What do you think? To test this
alternative explanation (i.e., it is due not to gender but to
education) you could examine the average income levels of
males and females ate each of the levels of education (i.e., to
see if males and females who have equal amounts of education
differ in income levels). If gender and income are still related
(i.e., if men earn more money than women at each level of
education) then you would conclude make this conclusion: “After
controlling for education, there is still a relationship between
gender and income.” And, by the way, that is exactly what
happens if you examine the real data (actually the relationship
becomes a little smaller but there is still a relationship). Can you
think of any additional variables you would like to control for?
That is, are there any other variables that you think will eliminate
the relationship between gender and income?

Other conditions…
 When attempting to establish condition 2,
researchers use logic and theory (e.g., we
know that biological sex occurs before
achievement on a math test) a design
approach that is used for this condition is a
longitudinal research because it establishes
proper time order.
 Condition 3 is a serious problem in
nonexperimental research because it is
always possible that an observed relationship
is "spurious" (i.e., due to some confounding
extraneous variable or "third variable").

Advantages of Correlational Methods
 Allow assessment of behavior as it occurs in people’s
everyday lives
 Allow study of variables that can’t be studied in
experimental designs (e.g. smoking, cancer)
 Establishes that a relationship between 2 variables exists
 One very serious disadvantage
 CORRELATION IS NOT CAUSATION!

Experimental Method
 Cornerstone of psychological research.
 Used to examine cause-and-effect relationships.
 Two essential characteristics:
 Researcher has control over the experimental procedures to make
sure that everything (but the variable being manipulated) stays the
same.
 Researcher manipulates one variable by changing its value from
one level to another. A second variable is observed (measured) to
determine whether the manipulation causes changes to occur
 Participants are randomly assigned to different treatment
conditions (they cannot self select).

Participant and Environmental Variables
 Participant Variables - things like
gender, age, and intelligence. Vary
from one individual to another
 Environmental Variables - lighting, time
of day, weather. Must be the same
across conditions

Random Sampling vs. Random Assignment
 Random Sampling
 Selecting Ps to be in
study so that everyone in
population has an equal
chance of being in the
study.
 Representative samples
 Generalization
 Is it possible?
 Random Assignment
 Assigning Ps (who are
already in study) to the
different conditions so
that each P as equal
chance of being in any of
the conditions.
 Equalizes the conditions
of experiment so that it is
unlikely that conditions
differ because of pre-
existing differences
 Required for inferences of
causality.

Variables
 Independent Variable
 variable that we expect causes an outcome
 the antecedent event
 variable that the experimenter can control and
manipulate
 Dependent Variable
 the “effect”
 the outcome variable
 it’s value depends on the changes introduced by
the IV

IVs and Conditions
 Must have at least two conditions (also called “levels”) of the
IV in order to demonstrate that the IV has an effect on the
DV. Otherwise, it wouldn’t be a ‘variable’.
 Experimental condition (IV present) vs. control condition (IV
not present)
 Those in control condition receive no treatment or receive neutral,
placebo treatment. Provides baseline for comparison with
experimental condition.
 Example
 interested in mood and helping
 experimental group – told they received “A” or “F”
 control group – does not grade feedback

Laboratory Experiments
 Conducted in settings in which:
 The environment can be controlled.
E.g., temperature, amount of light
in room
 The participants can be carefully
studied.
E.g., Ps remain in the same seat

Field Experiments
 Conducted in real-world settings.
 Advantage: People are more likely to
act naturally.
 Disadvantage: Experimenter has less control
(“quasi-experiments”).
 Quasi-independent variable - independent variable
in nonexperimental study. Typically something the
experimenter cannot manipulate such as gender
or smoking

Researchers are interested in influences on self-esteem.
Specifically, researchers want to assess how performing a
difficult task under pressure influences college students’
self-esteem. Ps are given a set of anagrams to solve. Half are
randomly assigned to receive very easy anagrams, and half are
given difficult ones. Crossed with this, half are randomly
assigned to be given 10 minutes to complete the anagrams,
and half are given 30 minutes to complete the task. After
completing as many of the anagrams as they can, Ps are given
a Q’aire labeled “Thoughts and Feelings
Questionnaire” that is really a measure of self-esteem.
 Constructs
 IV1: task difficulty
 IV2: pressure
 DV: self-esteem
 Operational
 IV1: easy vs. hard
 IV2: 10 vs. 30 minutes
 DV: score on Q’aire

Discrete and Continuous
Variables
 Discrete Variable - consists of separate, indivisible
categories. No values can exist between two
neighboring categories.
 One choice in a five point scale (1,2, 3, 4,5).
 Continuous Variable - infinite number of possible
values that fall between any two observed values.
Divisible into an infinite number of fractional parts
 Low Normal High
 Agree, strongly agree, disagree, strongly disagree

Scales of Measurement
 Nominal Scale
 Ordinal Scale
 Interval Scale
 Ratio Scale

Nominal Scales
 Nominal
Nominal scales are the lowest scales of measurement. Numbers are
assigned to categories as "names". Which number is assigned to which
category is completely arbitrary. Therefore, the only number
property of the nominal scale of measurement is identity. The number
gives us the identity of the category assigned. The only mathematical
operation we can perform with nominal data is to count.
Classifying people according to gender is a common application of a
nominal scale.
In the example below, the number "1" is assigned to "male" and the
number "2" is assigned to "female". We can just as easily assign the
number "1" to "female" and "2" to male. The purpose of the number is
merely to name the characteristic or give it "identity".


Nominal Scale
As we can see from the graphs, changing
the number assigned to "male" and
"female" does not have any impact on
the data -- we still have the same number
of men and women in the data set.

Ordinal Scale
Ordinal scales have the property of magnitude as
well as identity. The numbers represent a quality
being measured (identity) and can tell us whether a
case has more of the quality measured or less of
the quality measured than another case (magnitude).
The distance between scale points is not equal.
Ranked preferences are presented as an example of
ordinal scales encountered in everyday life. We also
address the concept of unequal distance between
scale points.

Coke, Pepsi or Sprite
 Ranked Preferences
We are often interested in preferences for different tastes, especially if we are
planning a party. Let's say that we asked the three students pictured below to rank
their preferences for four different sodas. We usually rank our strongest preference
as "1". With four sodas, our lowest preference would be "4". For each soda, we
assign a rank that tells us the order (magnitude) of the preference for that particular
soda (identity). The number simply tells us that we prefer one soda over another, not
"how much" more we prefer the soda.
Changing the numbers changes the meaning of the
preferences.

Interval Scales
 Interval scales have the properties of:
 identity
 magnitude
 equal distance
 The equal distance between scale points allows us to
know how many units greater than, or less than, one
case is from another on the measured characteristic.
So, we can always be confident that the meaning of
the distance between 25 and 35 is the same as the
distance between 65 and 75. Interval scales DO
NOT have a true zero point; the number "0" is
arbitrary.

IQ Scores are examples of an
Interval Scale

Ratio Scales
 Ratio scales of measurement have all of the
properties of the abstract number system.
 identity
 magnitude
 equal distance
 absolute/true zero
 These properties allow us to apply all of the possible
mathematical operations (addition, subtraction,
multiplication, and division) in data analysis. The
absolute/true zero allows us to know how many
times greater one case is than another. Scales with
an absolute zero and equal interval are

Where do t scores come in?
 We studied single variables such as
Central Tendency measures.
 However, most researchers look for
relationships between variables ( at
least two)
 The foundation to understanding the
relationship between them is the
correlation coefficient.

More than just one correlation
 The most widely used in education is
Pearson Product Moment correlation
coefficient.
 1) When do we use it?
 2) What does it tell us?
 1.We use coefficients to see how
variables are related to one another.
 2. We use Pearson PMC when the
variables being examined are ratio or
interval variables (continous variables)

Examples
 1. How much time in studying is related to
your exam scores?
 Assumptions for questions in your research:
 A. The more time spent in studying yields
higher scores in the exam.
 B. Students will do well in an exam if they
 Spend more time studying
 C. Some students will not do well in the exam
despite the hours spent studying.

Consider other possibilities
 The reason for studying longer hours is
because students do not understand
the material.
 Students will do well in the exam
regardless of the time spent studying.
Are the above exceptions to my rule?

How do we address this dilemma?
 Two fundamental characteristics of
correlations are:
 A) Direction- + --
 B) Strength r=.80 (is this low, moderate or
high
 Going back to our example, if student study
more, then what is the direction of the
correlation? What if they spend less time
studying?

What happens when…
 Students spend more time studying and the
scores go down. Why?
 The least time studying the scores go higher.
Why?
 So what is the direction in these cases and how
do we describe this in terms of the direction of
my correlation?
 Draw a simple scattergram showing each of the
above examples.

Strength (Magnitude) of Correlations
 Correlations may range from -1.0 to +1.0 . A
correlation of ) indicates no relationship.
 The closer the correlation to -1.0 or +1.0 the stronger
the relationship
 Perfect correlations are never found. Instead we find:
 -.20 and +.20=weak
 above 2.0 to .50=moderate
 Above .50 to .70= strong
 Some correlations despite their magnitude do not yield
any value. However, weak correlations may be of
great significance.

Examine the following Example
 r = is the symbol for correlations
Student Hours Spent
in Studying
(X)
Exam Score
(Y)
Student 1 5 80
Student 2 6 85
Student 3 7 70
Student 4 8 90
Student 5 9 85
*Scores must be paired because
you will convert scores to z scores
in order to subtract them from the
mean and dividing by the standard
deviation.

Consider the Following Scenarios
 One set of scores in variable x is associated
with high scores in variable y.
 One set of low scores in variable x is
associated with a set of low scores in variable y
 When two factors are positive then you get a
positive product.
 When two factors are negative you get a
positive product.
 Explain what these statements mean with

What happens when…
 High scores in one variable are associated with
low scores in the other variable?
Do not ascribe more to the association of the
variables!
 Correlation simply means that variations in the
scores of one variable correspond to the
variation of the scores of another variable.

Are correlations always linear?
A curvilinear relationship begin positive then it turns negative. There are some
explanations for this when it happens. Consider anxiety before a test and how it
may help to improve the scores. However, too much of it can cause negative
scores. This is alos a sign of a weak relationship.

Correlations can be calculated in
various ways:
X Values Y Values
5 80
6 85
7 70
8 90
9
85
X Value Y Value X*Y X*X(X2
) Y*Y (Y2
)
5 80 5 * 8 = 40 5 * 5 = 25
80 * 80 =
640
6 85 6 * 85 = 510 6 * 6 = 36
85 * 85 =
665
7 70 7 * 70 = 490 7 * 7 = 49
70 * 70 =
490
8 90 8 * 90 = 720 8 * 8 = 64 90 *90 = 810
9 85 9 * 85 =765 9 * 9 =81
85 * 85 =
665
Step 1: Find ΣX, ΣY, ΣXY, ΣX2
, ΣY2
.
ΣX =
ΣY =
ΣXY =
ΣX2
=
ΣY2
=
Step2: Now, Substitute in the above formula given.
Correlation(r) =[ NΣXY - (ΣX)(ΣY) / Sqrt([NΣX2
- (ΣX)2
][NΣY2
- (ΣY)2
])]

Review language for formulas
Hinton page
259

This is another way
This means:
•Calculate everybody's Z score.
•Multiply your Zx by your Zy.
•Add up these pairs for everyone.
•Divide by the number of people or observations.
So if we multiply your Z scores together, sum all these pairs and we get a positive
sum (positive scores for X multiplied by positive scores for Y and negative scores for
X multiplied by negative scores for Y) - we have a positivecorrelation)
If you have a negative relationship you will have thesumof positive Zxs times
negative Zys and vice versa. Add these up for a negative sum.
If you have no correlation, then you get equal numbers of positive Zx times
negative Zy, positive Zx times positive Zy, negative Zx times negative Zy, and
negative Zx times positive Zy. Add these up and you get zero.

Correlations may be used
when…
In the simple case of correlational research you have
one quantitative IV (e.g., level of motivation) and
one quantitative DV (performance on math test).
 · The researcher checks to see if the
observed correlation is statistically significant (i.e.,
not due to chance) using the "t-test for correlation
coefficients" (it tells you if the relationship is
statistically significant.
 · Remember that the commonly used
correlation coefficient (i.e., the Pearson correlation)
only detects linear relationships.

Let’s go back to our example
Student No hours spent
Studying
Exam Score
Joyce 0 95
Ashley 2 95
Jeff 4 100
Sean 7 95
Pedro 10 100
What can we conclude from the above examples?

We want to know…
1. Is there a correlation?
2. How strong and what the direction of
the correlation?
3. Is it statistically significant?
4. Is there a possibility of another
extraneous factor that gave us a false
correlation?

If you were testing the null
hypothesis…
 This where t test comes along.
 If your Ho= =0 (rho, population correlationƿ
coefficient)
 If your H1= 0ƿ≠
 The t distribution is used to test whether a
correlation coefficient is statistically significant.
 The t test just like z scores consist on a ratio where
t= r-ƿ
 sr

t tests
t= r-ƿ
sr
 r is the sample correlation coefficient
 is the population correlation coefficientǷ
 Sr is the standard error of the sample
correlation coefficient
 We do not need to calculate the standard
error of the sample because algebraic
equation allows us to use this formula:
2 2

Find a t value
 Go back to your formula where t=(.25) √ 100-2
 1-.25 2
 You can skip themath now and weget :
t =2.56, df= 98
Using appendix B we find that a t score of 2.56 has
a probability between 01 and .02 of occurring by
chance. If the researcher has used and alpha
value <.05 then the null hypothesis is rejected
and you can conclude that the correlation is
significant by accepting the alternative
hypothesis.
Thus r=.25 , t98,=2.56, < .05ƿ

You will learn more about alpha
values.
 The alpha value is the probability of making a Type I Error
. In a Hypothesis Test a Type I error occurs when
statistically unlikely test results lead to the incorrect
conclusion that the null hypothesis should be rejected. The
alpha value is conventionally designated by the symbol
alpha ( ).α
 The alpha value is also called the 'level of significance' and
is selected based on the importance of the test, a value of
0.05 would be common, with 0.01 for tests that are critical,
and even 0.001 in some cases.
 A hypothesis test involves comparing the calculated
p-value with the previously agreed alpha value. If the p-
value is less than the alpha value the alternative hypothesis
will be accepted.

The formula for calculating the
t value is
 t=(r) √ N-2
1-r2
Consider the following example: I randomly select 100 people living in different latitudes to measure whether the number of
hours expose to sunlight result in a seasonal affective disorder (SAD) measured by a mood scale of 1-10 where 1=“very sad”
and 10=“very happy”. Suppose that I have computed a Pearson Correlation wit my data and I find that there is the above
variables provided me with a .25 correlation. I want to know if this is statistically significant. So what do I do?

Summary: A picture is worth a
thousand words.

My Easy Way…
 ht
tp://easycalculation.com/statistics/corr
 http://easycalculation.com/statistics/le
 http://www.wadsworth.com/psychology_
d/templates/student_resources/worksho
ps/stat_workshp/scale/scale_08.html
 ht
tp://easycalculation.com/statistics/corr
 http://easycalculation.com/statistics/le
 http://www.wadsworth.com/psychology_
d/templates/student_resources/worksho
ps/stat_workshp/scale/scale_08.html

Correlations and t scores (2)

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