the nature of the data - relationships

The Nature of Your Data

The purpose of this presentation is to help you determine if the two data sets you are working with in this problem are:

The purpose of this presentation is to help you determine if the two data sets you are working with in this problem are:

Dichotomous by Scaled

Ordinal by Another Variable

Dichotomous by Dichotomous

Scaled by Scaled with at least one variable Skewed

First, let's define

what each of these mean.





Beginning with


What is dichotomous data?

The "Di" in Dichotomous means "two"

. . . and "tomous" or "tomy" as in “appendec-tomy” means to divide by.

So, dichotomous means to divide by two.

In this case a variable is divided by two or specifically it can only take on two values.

For example:

Gender is a good example of a dichotomous data.

Gender is a good example of a dichotomous data. It generally takes on two values

Gender is a good example of a dichotomous data. It generally takes on two values

(1) male

(2) female

In some cases individuals are divided by (1) those who received a treatment and (2) those who did not.

For example:

You have been asked to determine if those who eat asparagus score higher on a well-beingscale (1-10) than those who do not.

You have been asked to determine if those who eat asparagus score higher on a well-beingscale (1-10) than those who do not.

In this case, we are dealing with those (1) who eat asparagus and those (2) who do not.

With dichotomous by dichotomous data you are examining the relationship between two dichotomous variables.

Here is an example:

It has been purported that females prefer artichokes more than do males.


Dichotomous variable 1: Gender(1)Male(2)Female


Dichotomous variable 2: Artichoke Preference(1)Prefer Artichokes(2)Do not prefer Artichokes

Here is what the data set looks like:


Study Participant Gender1 = Male

2 = Female

Artichoke Preference1 = Prefer Artichokes

2 = Don’t Prefer Artichokes

A 1 2

B 2 1

C 1 2

D 2 1

E 2 1

F 1 2

G 1 2

This is an example of:


2 = Female



A 1 2

B 2 1

C 1 2

D 2 1

E 2 1

F 1 2

G 1 2

DichotomousData

This is an example of:


2 = Female



A 1 2

B 2 1

C 1 2

D 2 1

E 2 1

F 1 2

G 1 2

DichotomousData

byDichotomous

Data

As you will learn, there is a specific statistical method used to calculate the relationship between two dichotomous variables. It is called the Phi-coefficient.

Note - a dichotomous variable is also a nominal variable.

Note - a dichotomous variable is also a nominal variable. However, nominal variables can also take on more than two values:


1 = American

2 = Canadian

3 = Mexican

like so


1 = American

2 = Canadian

3 = Mexican

Dichotomous nominal variables can only take on two values - (e.g., 1 = Male, 2 = Female)

The next type of relationship involves dichotomous by scaled variables.

Now you already know what a dichotomous variable is, but what is a scaled variable?

A scaled variable is a variable that theoretically can take on an infinite amount of values.

For example,

Let's say a car can go as slow as 0 miles per hour and as fast as 130 miles per hour.

Within those two points (0 and 130mph) it could go 30 mph, 60 mph, 23 mph, 120 mph, 33.2mph, 44.302 mph, or even 88.00000000001 mph.

The point is that between these two points (0 and 130mph) there are an infinite number of values that the speed could take.

Scaled data also has what are called equal intervals.

Scaled data also has what are called equal intervals. This means that the basic unit of measurement (e.g., inches, miles per hour, pounds) are the same across the scale:

Scaled data also has what are called equal intervals. This means that the basic unit of measurement (e.g., inches, miles per hour, pounds) are the same across the scale:

40o - 41o

100o - 101o

70o - 71o

Each set of readings are the same distance apart: 1o

Slide 51

Here is an example of a word problem with scaled by dichotomous variables:

You have been asked to determine the relationship between age and hours of sleep. Age is divided into two groups: Middle Age (45-64) and Old Age (65-94).


The Scaled Variable is hours of sleep which can take on values

from 0 to 8+ hours.


The Dichotomous Variable is age which in this case can take on two values (1) middle and (2) old age.

Here is what the data set might look like:


Study Participant Age1 = 45-64 years2 = 65-94 years

Hours of Sleep

A 1 6.2

B 2 9.1

C 1 5.8

D 2 8.2

E 2 7.4

F 1 4.9

G 1 6.8



Hours of Sleep

A 1 6.2

B 2 9.1

C 1 5.8

D 2 8.2

E 2 7.4

F 1 4.9

G 1 6.8

DichotomousData



Hours of Sleep

A 1 6.2

B 2 9.1

C 1 5.8

D 2 8.2

E 2 7.4

F 1 4.9

G 1 6.8

DichotomousData

by



Hours of Sleep

A 1 6.2

B 2 9.1

C 1 5.8

D 2 8.2

E 2 7.4

F 1 4.9

G 1 6.8

DichotomousData

byScaled Data

Note, in the strictest sense scaled data should be like the car example (values are infinite between 0 and 130 mph).

However, in the social sciences many times data that is technically not scaled (e.g., on a scale of 1-10 how would you rate the ballerina's performance), are still treated as scaled data.

However, in the social sciences many times data that is technically not scaled (e.g., on a scale of 1-10 how would you rate the ballerina's performance), are still treated as scaled data.

Yes, it is true there are only 10 values that the variable can take on, but many researchers will treat it as scaled data. For the purposes of this class we will treat variables such as these as scaled data as well.

However, if we were rating on a scale of 1-2, 1-3or 1-4 we most likely would not treat such variables as scaled.

As you will learn there is a specific statistical method used to calculate the relationship between scaled by dichotomous variables. it is called the Point Biserial Correlation.

Next, let's consider the relationship involving ordinal data by another variable.

An ordinal variable is a variable where the numbers represent relative amounts of a an attribute. However, they do not have equal intervals.

For example,

In this pole vaulting example you will notice that 1st and 2nd place are closer to each other:


3rd

Place15’ 2”

2nd

Place18’ 1”

1st

Place18’ 3”


3rd

Place15’ 2”

2nd

Place18’ 1”

1st

Place18’ 3”

2 inches apart

. . . than 2nd and 3rd place, which are much further apart


3rd

Place15’ 2”

2nd

Place18’ 1”

1st

Place18’ 3”


3rd

Place15’ 2”

2nd

Place18’ 1”

1st

Place18’ 3”

3 feet 1” apart

Rank ordered or ordinal data such as these do not have equal intervals.

3rd

Place15’ 2”

2nd

Place18’ 1”

1st

Place18’ 3”

Here is what an ordinal by ordinal problem looks like:

In a study, researchers rank order different breeds of dog based on how high they can jump. They then rank order them based on the length of their hind legs. They wish to determine if a relationship exists between jumping height and hind leg length.

Here’s the data set:


Breed Participant Jumping Rank Hind-Leg LengthRank

A 1st 2nd

B 3rd 6th

C 6th 4th

D 4th 3rd

E 7th 7th

F 2nd 1st

G 5th 5th


Breed Participant Jumping Rank Hind-Leg LengthRank

A 1st 2nd

B 3rd 6th

C 6th 4th

D 4th 3rd

E 7th 7th

F 2nd 1st

G 5th 5th

Ordinal or Ranked Data


Breed Participant Jumping Rank Hind-Leg Length Rank

A 1st 2nd

B 3rd 6th

C 6th 4th

D 4th 3rd

E 7th 7th

F 2nd 1st

G 5th 5th

byOrdinal or

Ranked DataOrdinal or

Ranked Data

Rank ordered data can also take the form of percentiles.

Percentiles communicate the percentage of observations or values below a certain point.

If my score on the ACT is at the 35th percentile that means the 35% of ACT takers are below me.

A data set taken from the dog jumping question might look like this:


Breed Participant JumpingPercentile Rank

Hind-LegPercentile Rank

A 99% 85%

B 78% 33%

C 54% 64%

D 69% 73%

E 34% 28%

F 84% 97%

G 61% 54%




A 99% 85%

B 78% 33%

C 54% 64%

D 69% 73%

E 34% 28%

F 84% 97%

G 61% 54%

Ordinal or Percentile

Ranked Data




A 99% 85%

B 78% 33%

C 54% 64%

D 69% 73%

E 34% 28%

F 84% 97%

G 61% 54%

byOrdinal or Percentile

Ranked Data

Ordinal or Percentile

Ranked Data

The next example is that of a relationship between ordinal variable and a scaled variable.

You have been asked to determine if there is a relationship between the height of marathon runners and their final ranking in a race.


Marathon Runners Height in inches Order of Finish

A 73 6th

B 67 4th

C 69 5th

D 64 2nd

E 71 7th

F 62 1st

G 66 3rd



A 73 6th

B 67 4th

C 69 5th

D 64 2nd

E 71 7th

F 62 1st

G 66 3rd

ScaledData



A 73 6th

B 67 4th

C 69 5th

D 64 2nd

E 71 7th

F 62 1st

G 66 3rd

ScaledData

by Ordinal/Ranked Data

The final example is that of a relationship between ordinal variable and a nominal variable.

You have been asked to determine if there is a relationship between gender and spelling bee competition rankings.

Marathon Runners Gender Spelling Bee Rank

A 1 6th

B 2 4th

C 2 5th

D 2 2nd

E 1 7th

F 1 1st

G 2 3rd


A 1 6th

B 2 4th

C 2 5th

D 2 2nd

E 1 7th

F 1 1st

G 2 3rd

Dichotomous/Nominal Data


A 1 6th

B 2 4th

C 2 5th

D 2 2nd

E 1 7th

F 1 1st

G 2 3rd

by Ordinal/Ranked Data

Dichotomous/Nominal Data

In summary,

In summary, when at least one variable in the relationship is ordinal or rank ordered, then you choose the final option:

As you will learn there are specific statistical methods used to calculate the relationship between ordinal by ordinal or ordinal by other variables.

As you will learn there are specific statistical methods used to calculate the relationship between ordinal by ordinal or ordinal by other variables. They are the Spearman Rho and Kendall Tau.

As you will learn there are specific statistical methods used to calculate the relationship between ordinal by ordinal or ordinal by other variables. They are the Spearman Rho and Kendall Tau. We'll explain their difference in another presentation.

Lastly,

Lastly, let's consider the relationship involving scaled data with at least one variable having a skewed distribution.

For example,

You wish to determine the relationship between daily temperature and ice cream sales.

You wish to determine the relationship between daily temperature and ice cream sales .

MonthAverage Daily Temperature

Average Daily Ice Cream Sales

Jan 100

$100Feb 20

0$200

Mar 300

$300Apr 40

0$400

May 500

$500Jun 60

0$300

Jul 700

$200Aug 60

0$100

Sep 500

$300Oct 40

0$200

Nov 300

$400Dec 80

0$1000




Jan 100

$100Feb 20

0$200

Mar 300

$300Apr 40

0$400

May 500

$500Jun 60

0$300

Jul 700

$200Aug 60

0$100

Sep 500

$300Oct 40

0$200

Nov 300

$400Dec 80

0$1000

The skew is “0.00”

therefore temperature is

normally distributed




Jan 100

$100Feb 20

0$200

Mar 300

$300Apr 40

0$400

May 500

$500Jun 60

0$300

Jul 700

$200Aug 60

0$100

Sep 500

$300Oct 40

0$200

Nov 300

$400Dec 80

0$1000

The skew is “+3.23”

therefore ice cream sales is

Positively Skewed




Jan 100

$100Feb 20

0$200

Mar 300

$300Apr 40

0$400

May 500

$500Jun 60

0$300

Jul 700

$200Aug 60

0$100

Sep 500

$300Oct 40

0$200

Nov 300

$400Dec 80

0$1000




Jan 100

$100Feb 20

0$200

Mar 300

$300Apr 40

0$400

May 500

$500Jun 60

0$300

Jul 700

$200Aug 60

0$100

Sep 500

$300Oct 40

0$200

Nov 300

$400Dec 80

0$1000

This is an example where one variable is skewed and

the other normal

If your problem had one scaled variable that was skewed and the other normal or if both were skewed you would select:

A final note:

Dichotomous data like this:

1 = Catholic

2 = Mormon


1 = Catholic

2 = Mormon

StudyParticipants

Religious Affiliation

1 = Catholic2 = Mormon

A 1

B 1

C 1

D 2

E 1

F 2


1 = Catholic

2 = Mormon

. . . can become scaled if we are talking about the number of Catholics or Mormons.


1 = Catholic

2 = Mormon

. . . can become scaled if we are talking about the number of Catholics or Mormons.

Event Number of Catholics in attendance

Number of Mormons in attendance

A 120 22

B 322 34

C 401 78

D 73 55

E 80 3

F 392 102

Which option is most appropriate for the problem you are working with:

the nature of the data - relationships

Education