Psych 300A: Midterm 2 Review (Correlation)

Univariate data

Only one variable to describe, can be graphed (e.g. histogram) or numerically described (e.g. mean and SD)

Bivariate data

Data that has two variables, can be graphed on a scatterplot, can be numerically described with each individual variable and the association between the two variables can be described

Correlation

Statistical technique that is used to measure and describe the relationship between two variables, specifically how strongly and in what way the two variables are related

T or F: Knowing how two variables are correlated allows us to make predictions

T

What kind of variable is used with correlational data

Measured variable, typically one that cannot be manipulated

What are the 4 ways that the correlation can help us describe the data

1. Direction

2. Shape

3. Strength

4. Magnitude

Direction of correlation coefficient

Can be positive or negative

What does it mean if the correlation coefficient is negative, positive or zero

Negative - variables change in the opposite direction (one increases the other decreases)

Positive - variables change in the same direction (one increases the other increases)

Zero - no relationship, points are scattered widely

Shape of correlation coefficient

Can be linear or curvilinears

What does it mean if the shape of a scatterplot is linear or curvilinear

Linear - straight line relationship, data points are clustered around a line

Curvilinear - consistent predictable relationship that is expressed in a quadratic, cubic or quartic shape

What is the problem regarding the correlation coefficient if a scatterplot has a curvilinear relationship

It will underestimate the relationship as it is only meant to view linear relationships, this is why its important to always graph data

Strength of correlation coefficient

Subjective measure of relationship, can be weak moderate or strong depending on how closely the data points are clustered together

Magnitude of correlation coefficient

Objective measure of relationship based on computed r value that ranges from -1 to 1

T or F: an r value of -0.8 is weaker than an r value of +0.3  

F, it is stronger, the polarity simply explains the direction of the relationship

Pearson correlation

measures the degree and direction of the linear relationship between two variables

How is Pearson’s r calculated

Degree to which X and Y vary together / degree to which X and Y vary separately

What are the 3 steps to calculate Pearson’s r

1. Plot data

2. Compute univariate stats (mean and SD for each variable separately)

3. Compute bivariate stats (compute the relationship between deviation scores to determine if they deviate in the same or opposite direction)

How does one calculate bivariate stats

HINT: 4 steps

1. Compute deviation scores

2. Compute sums of products (SP)

3. Compute covariance (COV)

4. Compute Correlation coefficient (r)

How does one compute deviation scores

Scores of X = x - x̄ 

Scores of Y = y -ȳ

Sums of Products (SP)

Tells us whether the scores deviate in the same or opposite direction (essentially the same as SS, but with two variables)

How is SP calculated

SP = Σ(x−x̄)(y−ȳ)

Covariance (COV)

Measure of the average extent for which scores on two variables covary from their respective means across the entire group of scores

How is COV calculated

COV = SP / N = Σ(x−x̄)(y−ȳ) / N

What does COV tell us if the number is positive, negative or zero

Positive - Deviate consistently in the same direction, graph is positive

Negative - Deviate consistently in the opposite direction, graph is negative

Zero - Some values vary in the same direction others vary in the opposite direction

How is Pearson’s r calculated

r = COV_xy / SD_xSD_y

What are the 2 major categories for interpreting the Pearson correlation

1. Human behaviour

2. Test reliability

What are the general guidelines for human behaviour

No relationship - value is between 0 and |.10|

Weak relationship - value ranges from |.10| to |.30| 

Moderate - value ranges from >|.30| to |.50|

Strong - value is > |.50|

What are the general guidelines for test reliability

Very desirable - value is > .85

Moderately desirable - value is .70 to .85 (moderately acceptable)

Not desirable - value is < .70 (poor reliability)

Note: Test reliability must be positive

What are the 6 major factors effecting r

1. Sampling error

2. Unmeasured 3rd variable

3. Heterogenous sample

4. Sampling from a truncated range

5. Non-linearity

6. Heteroscedasticity in data

Sampling error

Naturally occurring discrepancy that exists between a sample statistic and corresponding parameter (sample and parameter will never be equivalent), can make r larger or smaller

Unmeasured third variable

Could be that correlation is caused by an outside variable, why we cannot assume causation from correlation, can cause r to be larger or smaller

Mediator

Type of unmeasured third variable, when an outside variable exists due to a cause that effects the outcome (cause does not directly imply outcome, mediator explains this)

Moderator

Type of unmeasured third variable that can affect the relationship in an unknown way by influencing the strength of the relationship

Spurious

Type of unmeasured third variable, there is no cause and simply a correlation due to a number of outside forces

Heterogenous sample

Data in which the sample of observations can be subdivided into two distinct sets on the basis of some other variable, can cause r to be larger or smaller

T or F: a heterogenous sample can cause r to be larger, smaller or even zero

T, can imply a relationship that does not exist, can also cause r = 0 due to the two homogenous groups being oppositely correlated from one another

Sampling from truncated range

Severely restricted range may provide different correlation, can cause r to be larger or smaller

T or F: We an generalize a correlation beyond the sample range of data

F, can lead to a correlation that is entirely false

Non-linearity

We may get a smaller r value due to the data being non-linear, can lead to r being underestimated

Heteroscedacticity in data

Variance in y is not constant across the range of x variables, typically caused by a skew in one/both variables, can cause r to be underestimated

Homoscedasticity

Variable in Y scores remains constant across increasing values of X (this is a good thing)

What are the 4 types of correlations we calculate in this class

1. Pearson’s r

2. Spearman rho (ρ, r_s)

3. Point biserial (r_pb)

4. Phi (ɸ)

What kind of variables need to be used with Pearson’s r

Both variables need to be interval/ratio

What kind of variables need to be used with spearman rho

used any time at least one of our variables are ordinal and the other is interval/ratio/nominal or if there is a weak curvilinear relationship with interval/ratio data

T or F: spearman rho should be used if data heteroskedastic

T

How is spearman rho calculated

Convert all scores into ranks and then use the pearson formula to find how consistent increase in one variable are associated with another

T or F: if pearson r is 1 then spearman rho is also 1

T

Point biserial

Used when one variable is nominal and the other is interval/ratio, calculated using the pearson formula (only if nominal value is ≤ 2 categories)

Phi

Use when both variables are nominal, both groups must be dichotomous

Dichotomous

Categorical variables with only two possible, mutually exclusive outcomes (e.g. yes/no)