stats chapter 4

Chapter 4

More about relationships between 2 variables

4.1 TRANSFORMING TO ACHIEVE LINEARITY

What if the scatterplot is not linear?

• Of course not all data is linear!• Our method in statistics will involving

mathematically operating on one or both of the explanatory and response variables

• An inverse transformation will be used to create a non-linear regression model

• This will be a little “mathy”

Transformations

• Before we begin transformations, remember that some well known phenomenon act in predictable ways– I.e. when working with time and gravity,

you should know that there is a square relationship between distance and time!

The Basics

• The data from measurements (raw data) must be operated on.

• Apply the same mathematical transformation on the raw data– Ex. “Square every response”

• Use methods from the previous chapter to find the LSRL for the transformed data

• Analyze your regression to ensure the LSRL is appropriate

• Apply an inverse transformation on the LSRL to find the regression for the raw data.

Example

Please refer to p 265 exercise 4.2Length (cm) Period (s)

16.5 0.777

17.5 0.839

19.5 0.912

22.5 0.878

28.5 1.004

31.5 1.087

34.5 1.129

37.5 1.111

43.5 1.290

46.5 1.371

106.5 2.115

Example

• Data inputted into L1 and L2

• Scatterplot• Looks pretty good,

right?

Exercise

• LSRL• Y=.6+.015X

r = 0.991• Residual Plot• Perhaps we can do

better!

Example

• L3 = L2^.5 (square root)

• LinReg L1, L3• Note that the value

of r2 has increased• Note that the value

of the residual of the last point has decreased

Exponential Models

• Many natural phenomenon are explained by an exponential model.

• Exponential models are marked by sharp increases in growth and decay.

• Basic model: y = A·Bx

• For this transformation, you need to take the logarithm of the response data.

• You may use “log10” or “ln” your choice.– I prefer “ln” (of course)

Exponential Models

After the transformation, we have the following linear model: ln(y) = a + b·x

1. ln(y) = a + b·x2. eln(y) = e(a + b·x) exponentiate3. y = ea · ebx property of

logarithms4. Let ‘A’ = ea redefine variables

‘B’ = eb

5. y = A·Bx this is our model

Exponential Models

• Since this is an ‘applied math’ course, you need not remember how to apply the inverse transformation

• Whew• BUT you do need to memorize:

when ln(y) = a + bxy = A·Bx

where ‘A’ = ea and ‘B’ = eb

Exponential Models

Let’s try this data

Exponential Models

Take the ln of L2- the response list and store in

L3

Exponential Models

These are our “transformed responses”

Exponential Models

From our homescreen, we perform an LSRL

using the transformed data

Exponential Models

We don’t have to store this regression for transformed

data

Exponential Models

Take note of the values of ‘a’ and ‘b’

Exponential Models

A quick look at the residuals

Exponential Models

The values of the residuals are small .. . no defined pattern

Exponential Models

• Our regression model is exponential y = A·Bx

Where A = ea and B = eb • y = e0.701 · (e0.184)x

Exponential Models


Where A = ea and B = eb • y = e0.701 x (e0.184)x

Exponential Models


Where A = ea and B = eb • y = e0.701 x (e0.184)x

• Ory = 2.06 · (1.20)x

Exponential Models

Put our regression in Y1

Exponential Models

Change Plot1 from a resid. to a scatter plot

Exponential Models

Looks pretty good, eh?

Power Models

• These models are used when the rate of increase is less severe than an exponential model, or if you suspect a ‘root’ model

• For this model, you will find the logarithms of both the expl var and the resp var

Power models

LSRL on transformed data yields:ln(y) = a + b·ln(x)

1. ln(y) = a + b·ln(x)2. e ln(y) = e(a + b·ln(x))

3. y = ea·eln(x^b)

4. y = ea ·xb

5. Let ‘A’ = ea

6. y = A · xb

Power models

Let’s use this data to find a power model

Power models

This time we need to transform both lists

Power models

Transformed exp = L3Transformed resp = L4

Power models

LSRL on transformed datano need to store in Y1

Power models

Take note of the values of ‘a’ and ‘b’

Power models

A quick look at the residuals

Power models

Note that we use the transformed exp var

Power models

No defined pattern

Power models

Residuals are all small in size

Power models

• When ln(y) = a + b·ln(x),y = A · xb

where ‘A’ = ea

Our model is y = (e1.31)· x1.27

Power models


where ‘A’ = ea

Our model is y = (e1.31) · x1.27

Power models


where ‘A’ = ea

Our model is y = (e1.31) · x1.27

Or y = 3.71 · x1.27

Power models

Regression in Y1

Power models

Change from resid to scatter plot

Power models

(notice L1 and L2)

Power models

Looks pretty good!

Power models

• Much like the exponential model, you only need to know how the transformed model becomes the model for the raw data.


where ‘A’ = ea

Transformation thoughts

• Although this is not a major topic for the course, you still need to be able to apply these two transformations (exp and power)

• Be sure to check the residuals for the LSRL on transformed data! You may have picked the wrong model :/

• If one model doesn’t work, try the other. I would start with the exponential model.

• Don’t transform into a cockroach. Ask Kafka!

Assn 4.1

• pg 276 #5, 8, 9, 11, 12

4.2 RELATIONSHIPS BETWEEN CATEGORICAL VARIABLES

Marginal Distributions

• Tables that relate two categorical variables are called “Two-Way Tables”– Ex 4.11 pg 292

• Marginal Distribution– Very fancy term for “row totals and column

totals”– Named because the totals appear in the

margins of the table. Wow.

• Often, the percentage of the row or column table is very informative


Age Group

Female

Male Total

15-17 89 61 15018-24 5668 4697 1036525-34 1904 1589 349435 or older

1660 970 2630

Totals 9321 7317 16639


Age Group

Female

Male Total

15-17 89 61 15018-24 5668 4697 1036525-34 1904 1589 349435 or older

1660 970 2630

Totals 9321 7317 16639

Column Totals


Age Group

Female

Male Total

15-17 89 61 15018-24 5668 4697 1036525-34 1904 1589 349435 or older

1660 970 2630

Totals 9321 7317 16639

Row Totals


Age Group

Female

Male Total

15-17 89 61 15018-24 5668 4697 1036525-34 1904 1589 349435 or older

1660 970 2630

Totals 9321 7317 16639

Grand Total

Marginal Distributions “Age Group”


Age Group

Female

Male Total Marg. Dist.

15-17 89 61 15018-24 5668 4697 1036525-34 1904 1589 349435 or older

1660 970 2630

Totals 9321 7317 16639


Age Group

Female


15-17 89 61 15018-24 5668 4697 1036525-34 1904 1589 349435 or older

1660 970 2630

Totals 9321 7317 16639Row total / grand

total150/16639=0.009


Age Group

Female


15-17 89 61 150 0.9%18-24 5668 4697 1036525-34 1904 1589 349435 or older

1660 970 2630

Totals 9321 7317 16639Row total / grand

total150/16639=0.009


Age Group

Female


15-17 89 61 150 0.9%18-24 5668 4697 10365 62.3%25-34 1904 1589 3494 21.0%35 or older

1660 970 2630 15.8%

Totals 9321 7317 16639 100%

Adds to 100%

Marginal Distributions “Gender”

Age Group

Female

Male Total

15-17 89 61 15018-24 5668 4697 1036525-34 1904 1589 3494

35 &up 1660 970 2630Totals 9321 7317 16639Margin

dist.56% 44% 100%

Similarly for columns

Describing Relationships

• Some relationships are easier to see when we look at the proportions within each group

• These distributions are called “Conditional Distributions”

• To find a conditional distribution, find each percentage of the row or column total.

• Let’s look at the same table, and find the conditional distribution of gender, given each age group

Conditional DistributionsAge

GroupFemale Male Total

15-17 89 61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)



15-17 89 61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

We will look at the conditional

distribution for this row



15-17 89 61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

This cell is 89/150 (cell total /row total)

=53.9%



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)


=59.3%



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)


=40.7%



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

The table with complete

conditional distributions for

each row



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

For an analysis of the effect of age

groups, compare a row’s conditional

distribution…



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

With the marginal distribution for the

columns…



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

They should be close …



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

… unless there is an effect caused by

the age group (?)



15-17 89(59.3%)

61(40.7%)

150(100%)

18-24 5668(54.7%)

4697(45.3%)

10365(100%)

25-34 1904(54.5%)

1589(45.5%)

3494(100%)

35 or older

1660(63.1%)

970(36.9%)

2630(100%)

Totals 9321(56%)

7317(44%)

16639(100%)

… and these are not close to the

marginal distribution!

Conditional Distributions

• Based on the previous table, the distribution of “gender given age group” are not that different.

• We can see that the “35 and older” group seems to differ slightly from the overall trend.

Conditional Distributions “age group given gender”

Age Group

Female Male Total

15-17 89(1%)

61(0.8%)

150(0.9%)

18-24 5668(60.8%)

4697(64.2%)

10365(62.3%)

25-34 1904(20.4%)

1589(21.7%)

3494(21.0%)

35 or older

1660(17.8%)

970(13.3%)

2630(15.8%)

Totals 9321(100%)

7317(100%)

16639(100%)


Age Group

Female Male Total

15-17 89(1%)

61(0.8%)

150(0.9%)

18-24 5668(60.8%)

4697(64.2%)

10365(62.3%)

25-34 1904(20.4%)

1589(21.7%)

3494(21.0%)

35 or older

1660(17.8%)

970(13.3%)

2630(15.8%)

Totals 9321(100%)

7317(100%)

16639(100%)

Here is the same chart with the

conditional distributions by

gender…


Age Group

Female Male Total

15-17 89(1%)

61(0.8%)

150(0.9%)

18-24 5668(60.8%)

4697(64.2%)

10365(62.3%)

25-34 1904(20.4%)

1589(21.7%)

3494(21.0%)

35 or older

1660(17.8%)

970(13.3%)

2630(15.8%)

Totals 9321(100%)

7317(100%)

16639(100%)

Is there a gender effect noticeable from this table?


Age Group

Female Male Total

15-17 89(1%)

61(0.8%)

150(0.9%)

18-24 5668(60.8%)

4697(64.2%)

10365(62.3%)

25-34 1904(20.4%)

1589(21.7%)

3494(21.0%)

35 or older

1660(17.8%)

970(13.3%)

2630(15.8%)

Totals 9321(100%)

7317(100%)

16639(100%)

Conditional Distribution

Conclusions from the previous chart• Females are more likely to be in the “35

and older group” and less likely to be in the “18 to 24” group

• Males are more likely to be in the “18 to 24” group and less likely to be in the “35 and older” group

• These differences appear slight. Are actually “significant” with respect to the overall distribution?

Conditional Distribution

• No single graph portrays the form of the relationship between categorical variables.

• No single numerical measure (such as correlation) summarizes the strength of the association.

Simpson’s Paradox

• Associations that hold true for all of several groups can reverse direction when teh data is combined to form a single group.

• EX 4.15 pg 299• This phenomenon is often the result

of an “unaccounted” variable.

Assignment 4.2

• Pg 298 #23-25, 29, 31-35

4.3 ESTABLISHING CAUSATION

Different Relationships

• Suppose two variables (X and Y) have some correlation– i.e. when X increases in value, Y

increases as well– One of the following relationships may

hold.


Causation• In this relationship, the explanatory

variable is somehow affecting the response variable.

• In most instances, we are looking to find evidence of a causation relationship


Causation


Common Response• In this relationship, both X and Y are

correlated to a third (unknown) variable (Z).

• EX, When Z increases X increases and Y increases.

• Unless we known about Z, it appears as though X and Y have a causation relationship.


Common Response


Confounding• X and Y have correlation, • An (often unknown) third variable ‘Z”

also has correlation with Y• Is X the explanatory variable, or is Z

the explanatory variable, or are the both explanatory variables?


Confounding

Causation

• The best way to establish causation is with a carefully designed experiment– Possible ‘lurking variables’ are controlled

• Experiments cannot always be conducted–Many times, they are costly or even

unethical

• Some guidelines need to be established in cases where an observational study is the only method to measure variables.

Causation- some criteria

• Association is strong• Association is consistent (among

different studies)• Large values of the response variable

are associated with stronger responses (typo?)

• The alleged cause precedes the effect in time

• The alleged cause is probable

Assignment 4.3

Pg312 #41, 45, 50, 51

Chapter 4 Review

• #37, 53, 54, 57

stats chapter 4

Education

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