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What does it means when we say that there is a correlation between two variables?
It indicates that there is a systematic relationship between those variables.
1. As one variable changes, the other variable would be expected to systematically increase or decrease to some degree.
Example: high
rate of attending class, they tend to earn higher grades on exams.
Please note that when we calculate correlations, scores are
paired together (much like they are for a paired-samples t
test).
Correlation is not causation.
Linear Relationship
Relationships that can be described by a straight line (rather than a non-linear function, such as curve).
Correlation Coefficient
A statistic that quantifies the relation between two variables.
Properties of correlation coefficient
1. The correlation coefficient can be positive or negative.
2. The correlation always has a value that is between (or
equal to) -1.00 and +1.00
3. The magnitude of the coefficient indicates how strong the correlation is, not the sign of the
coefficient.
3.1 Coefficients of +0.80 and -0.80 both indicate equally strong correlations.
3.2 As a coefficient becomes more distant from 0.00, it indicates a stronger correlation.
Positive correlation
High scores tend to be paired with high scores (across the two variables), and low scores tend to be paired
with low scores.
Negative Correlation
High scores tend to be paired with low scores (across the two variables), and low scores tend to be paired with high scores.
What are some examples of positive vs. negative Correlations?
A thermometer; as temperature increases, the mercury level rises (higher temperatures are paired with higher mercury levels).
A teeter-totter (Sea saw); as one side of it
rises, the other side drops.
(+): variables increase
or decrease togethe
Positive correlation
(-): One variable
increases, while other decreases
Negative Correlation
Size (strength) of the Correlation
The sign indicate the direction of the
relationship between variables. The magnitude indicates the strength or size of the correlation.
Small: 0.10
Medium: 0.30
Large: 0.50
What does it means when r = 0?
No relationship between two variables.
Imagine that every data point falls perfectly on a diagonal line of best fit in a scatter plot. What is the absolute value of the corresponding Pearson correlation coefficient (r)?
r = 1.00 or -1.00
Perfect linear Correlations.
Mallory found a large (strong) correlation between the amount of time people spent studying and the grades they received on a test. In her sample, people who studied less performed worse on the test. The correlation coefficient in this study was most likely ___________.
a.+1.50 b.+0.25 c. -0.80 d.+0.75
d. r = +0.75
Correlation is not the same as Causation.
Just because Variable A is correlated with Variable B, it does not necessarily mean that Variable A causes
Variable B (or vice versa).
Did taking your umbrella with you (or not taking it)
cause it to rain?
Ways for variables to be correlated
A third
variable (variable C) might cause both variable A and B.
Note: there might be many "third" variables, and we may not be able to identify them all.
How do we identify all the variables in an experiment? Causation or correlation?
Though a good internal validity experiment we can distinguish between these possibilities: causation or correlation.
The Pearson
Correlation Coefficient
Or,
the Pearson product moment correlation coefficient
It is a statistic that quantifies the linear relationship between two scale variables.
It is not appropriate for
describing non-linear relationships. To appropriately describe a relationship with the Pearson correlation
coefficient, you need a relationship that can be described by a straight line.
What is the pearson correlation coefficient denoted by when working with a sample and a population?
Sample: r.
Population: "ρ" (pronounced "rho")
Would r be able to describe a Non-linear relationship?
No. r would not be appropriate for describing this relationship. (Nonetheless, a relationship is clearly present).
"Sex" and "SSy" represent...
(X-Mx)^2 and (Y-My)^2.
The sum of squared deviations for each score from it's respective mean, for each variable (i.e., Variable X and Variable Y).
X and Y mean...
Single sample values
Mx and My
Mean of x or Y.
Calculating r
1. Separate x and y. then Calculate the mean for both.
2. calculate (x-Mx) and (y-My).
3. calculate the Sum of [(x-Mx) * (y-My)]
4. calculate the sum of SSx and SSy.
5. plug everything into the equation.
6. calculate r.
Assumptions and Important ideas when Using r
1. Data are randomly selected from the population.
2. The underlying population
distributions for our variables are approximately normal.
3. Homoscedasticity.
4. Other important ideas:
We have scale variables (interval or ratio) when computing r. We are looking at a linear relationship. Outliers can have a strong impact on the value of r. We have pairs of scores.
Homoscedasticity
Each variable should vary equally across all levels of the other variable. (For example, we should not have more variability in grades at high levels of absenteeism vs. low levels of absenteeism.
Steps of Hypothesis Testing
1. identify the population, distribution, and assumptions, and then choose the appropriate hypothesis test.
2. state the null and research hypotheses, in both words and symbolic notations.
3. determine the characteristics of the comparison distribution.
4. determine the critical
values or cutoffs, that indicate the points beyond which we will reject the null hypotheses.
5. calculate the test statistic.
6. decide whether to reject or fail to reject the null hypothesis.
Hypothesis testing step 2: State the null and alternative hypotheses in words and symbolic notation.
• Null hypothesis: There is no correlation between the number of
absences and exam grades.
• Alternative
hypothesis: There is a correlation between the
number of absences and exam grades.
• H0: ρ= 0
• H1: ρ≠ 0
• Note that our hypotheses are about the population (even
though we have samples).
• In this case, we are also conducting a two-tailed test.
Hypothesis step 3: Determine the characteristics of the comparison distribution.
df = N-2 (where N is the number of pairs of scores).
Decide to reject or fail to reject the null hypothesis.
Same as any type of testing. If the calculated value is more extreme than our critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.
The Importance of Testing for Statistical Significance and the Influence of Outliers
An outliner might mislead us to
think that there is a strong correlation when the correlation is not statistically significant.
Sometimes, even large sample size give us an estimate that is much closer to the true correlation of 0; However, the correlation is not statically significant.
p value on SPSS output for correlations
Sig. (2 tailed).
What is the alternative approach for calculating r?
By using covariance
covariance
It represents the degree to which two variables vary together.
EX:
1. When high scores on one variable are paired with high scores on the other, the covariance will be positive and large.
2. When high scores on one variable are paired with low scores on the other, the covariance will be negative and large.
What is N in the covariance formula?
Number of pairs of scores.
Additional information: covariance and correlation
Similar to the other calculation for r, in addition, it adds divide by (N-1) to standard deviation of x and y. It(N-1) includes in covariance equation.
Why is r a scaled (standardized) version of the covariance.
1. Covariance divided by the product of the standard deviations of x and y.
2. This scaling restricts the values of r such that it must be between -1
and +1 (inclusive). Again, the coefficient r represents the strength of a linear relationship between two scale variables.
Test-retest Reliability
It refers to whether the scale being used provides consistent information every time the test is taken.
Internal Reliability
[consistency] between items in the same scale.
Are all items in a scale measuring the same idea?
Split-half reliability
correlate answers to odd and even items in a scale.
Coefficient alpha
it is a commonly used estimate of internal reliability. can be thought of as an average of all possible split-half correlations.
Criterion validity
look for correlations between your measure and real-world outcomes.
Convergent Validity
Look for correlations between your measure and other, similar measures.
Discriminant Validity
Look for a lack of correlation between your measure and dissimilar measures.
Reliability and Validity
when a measure is reliable, that doesn't mean that it is also valid.
Valid
Accurate.
It is
measure what it is suppose to measure.
Partial Correlations
Degree of association between two variables after statistically removing the association of a their variable.
What does partial correlations allow a researcher to do?
It allows a researcher to "control for" the presence of a third variable.
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