Math 15Introduction to Scientific Data Analysis

Lecture 5Association Statistics & Regression Analysis

University of California, Merced

Week Date Concepts Project Due

1

2 January 28 Introduction to the data analysis

3 February 4 Excel #1 – General Techniques

4 February 11 Excel #2 – Plotting Graphs/Charts Quiz #1

5 February 18 Holiday

6 February 25 Excel #3 – Statistical Analysis Quiz #2

7 March 3 Excel #4 – Regression Analysis

8 March 10 Excel #5 – Interactive Programming Quiz #3

9 March 17 Introduction to Computer Programming - Part - I

March 24 Spring Recesses

10 March 31 Introduction to Computer Programming - Part - II Project #1

11 April 7 Programming – #1 Quiz #4

12 April 14 Programming – #2

13 April 21 Programming – #3 Quiz #5

14 April 28 Programming – #4

15 May 5 Programming - #5 Quiz #6

16 May 12 Movies / Evaluations Project #2

Final May ??? Final Examination

Course Lecture Schedule

Quiz Next Week!

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Project #1 – Due March 31st, 2008

Projects can be performed individually or in groups of three, with following rules: Teams turn in one project report and get the same grade. A team consists of at most 3 people—no copying between

teams! Team project report must include a title page, where a team

describe each team member’s contribution. 10% bonus for projects done individually Individual projects must not be copied from anyone else No late project will be accepted!

Project #1 will be posted at UCMCROP by Next Monday!

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Review:Measures of dispersion or variability

Variance or Standard Deviation The one on the left is more dispersed than the one

on the right. It has a higher variance or standard deviation.

Average

Mode

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Which is more precise measurement?

Although the standard deviation is a good measure of the precision of a given set of data, it can be difficult to compare the standard deviation from two different types of measurements directly.

You might need to do such a comparison to determine the largest source of uncertainty in an experimentally determined answer

446 35.49

Average

mg ml

s (standard Deviation)= 23

s = 4.5

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Get the Right Tool for the Job!

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Measures of dispersion or variability

One way to do this comparison A relative standard deviation, RSD, is simply the ratio of the

standard deviation over the mean

446 35.49

Average

mg ml

s = 23

s = 4.5

xRSD

100

RSD = 100x(23/446) = 5.2

RSD = 100x(4.5/35.49) = 12.7

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Any Questions?

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Common Practice for Data Analysis

A common task in data analysis is to investigate an association between two variables. To see if two variables vary together

To see how one variable affect another.

Correlation

Regression

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Correlation

A correlation tells us whether the two variables vary together. i.e. as one goes up the other goes up

(or goes down)

Correlation Coefficient(Pearson product-moment correlation coefficient or Pearson’s

r)

Correlation Coefficient(Pearson product-moment correlation coefficient or Pearson’s

r)

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Correlation Coefficient

Vary from +1 (perfect correlation) through 0 (no correlation) to -1 (perfect negative correlation)

sales A

9

12

15

18

0 1 2 3 4 5 6 7

day

sa

les

Series 4

0

10

20

30

0 2 4 6 8 10 12Series 1

0

5

10

15

20

0 2 4 6 8 10 12

1r 1r

0r

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Correlation Coefficient – cont.

Always draw a diagram to check There are no OUTLIERS. If there are outliers,

the following may not apply. The relation is not curved (r only refers to

LINEAR correlation)

r (approx.

)

strength of tendency

what with what

0.9 to 1 strong high y with high x and low y with low x

0.7 to 0.9 some high y with high x and low y with low x

0.3 to 0.7 little high y with high x and low y with low x

-0.3 to 0.3

none neither high nor low y with high or low x

-0.3 to -0.7

little low y with high x and high y with low x

-0.7 to -0.9

some low y with high x and high y with low x

-0.9 to -1 strong low y with high x and high y with low x

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Excel Function – Correlation Coefficient

= CORREL(array1,array2)or

= PEARSON(array1,array2)

Positive Correlation

Lengths of a leg bone (in cm) in penguin mating pairs

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Ice cream sales vs. number of people who drown at sea

Month# of Ice cream sales

(in million)# of

Drowning1 0.30 02 0.20 03 0.90 14 1.50 15 2.00 36 3.50 57 5.50 68 8.00 89 7.50 5

10 2.50 111 0.80 012 0.70 0

Correlation Coefficient 0.927

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Wait!

What kinds of conclusion can we make from the correlation relationship?

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Examples

Ice cream sales correlate with the number of people who drown at sea. Therefore, ice cream causes people to

drown.

Since the 1950s, both the atmospheric CO2 level and crime levels have increased sharply. Hence, atmospheric CO2 causes crime.

Not Good Ones!

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Ice cream sales vs. number of people who drown at sea

Month# of Ice cream sales

(in million)# of

Drowning1 0.30 02 0.20 03 0.90 14 1.50 15 2.00 36 3.50 57 5.50 68 8.00 89 7.50 5

10 2.50 111 0.80 012 0.70 0

Correlation Coefficient 0.927

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Correlation does not imply causation

There can be no conclusion made regarding the existence or the direction of a cause and effect relationship only from the fact that A is correlated with B. Correlation Coefficient only tells you whether the

two variables vary together.

Determining whether there is an actual cause and effect relationship requires further investigation, even when the relationship between A and B is statistically significant, a large effect size is observed, or a large part of the variance is explained.

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Any Questions?

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Regression

Regression is used when we have some reasons to believe that changes in one variable cause changes in the other. Correlation coefficient is not evidence for a causal

relationship.

The simplest kind of causal relationship is a straight-line (or linear) relationship.

Linear regression Linear regression

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Linear regression

Linear regression assumes a linear relationship between two variables: Dependent factor, y, and independent factor, x.

In a mathematical approach, this relationship can be described by the following linear equation:

where a is called the slope and b is called the intercept. This equation, which allows you to calculate y

(dependent) based on x (independent), is based on the least square method.

baxy

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Review - Math

Linear Equation Slope and Intercept

y20 y = ax + b

a1

10 a1

b

0 1 2 3 4 x

8

3

y = 3x + 8

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Slope & Intercept formula

Pair Female Male1 17.1 16.52 18.5 17.43 19.7 17.34 16.2 16.85 21.3 19.56 19.6 18.3

Slope 0.5205Intercept 7.8830

Pair Female Male1 17.1 16.52 18.5 17.43 19.7 17.34 16.2 16.85 21.3 19.56 19.6 18.3

Slope =SLOPE(C2:C7,B2:B7)Intercept =INTERCEPT(C2:C7,B2:B7)

Y-values

X-values

Lengths of a leg bone (in cm) in penguin mating pairs

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y = ax + b

a – slope & b - intercept

Pair Female Male1 17.1 16.52 18.5 17.43 19.7 17.34 16.2 16.85 21.3 19.56 19.6 18.3

Slope 0.5205Intercept 7.8830

Pair Female Male Predicted Male1 17.1 16.5 16.782 18.5 17.4 17.513 19.7 17.3 18.144 16.2 16.8 16.315 21.3 19.5 18.976 19.6 18.3 18.08

Slope 0.5205Intercept 7.8830

X-values

Predicted Y-values

=$C$10*B3+$C$11

123456789

101112

B C

X-value

Don’t forget $ sign!

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Plot a linear regression (or trend) line – Part 1

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Female size (mm)

Mal

e si

ze (

mm

)

Pair Female Male Predicted Male1 17.1 16.5 16.782 18.5 17.4 17.513 19.7 17.3 18.144 16.2 16.8 16.315 21.3 19.5 18.976 19.6 18.3 18.08

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Female size (mm)

Mal

e si

ze (

mm

)

You can add a linear regression line

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Plot a linear regression (or trend) line –Part 2

Pair Female Male1 17.1 16.52 18.5 17.43 19.7 17.34 16.2 16.85 21.3 19.56 19.6 18.3

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Female size (mm)

Mal

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mm

)

Right-click on any data point on the graph Choose Add Trendline Click on Options tab, and select Display

equation and Display R-squared. Click “Ok”

Don’t forget to check these two parts!

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Plot a linear regression (or trend) line –Part 2 – cont.

R2 Value (R-squared value – RSQ) “measure of scatter”

The closer this value comes to 1, the more accurate the prediction.

y = 0.5205x + 7.883

R2 = 0.7767

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Female size (mm)

Mal

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mm

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Let’s review the process!

Pair Female Male1 17.1 16.52 18.5 17.43 19.7 17.34 16.2 16.85 21.3 19.56 19.6 18.3

Lengths of a leg bone (in cm) in penguin mating pairs

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Female size (mm)

Mal

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mm

)

If there are some reasons to believe some causalities between two variables, then, plot a graph!

Pair Female Male1 17.1 16.52 18.5 17.43 19.7 17.34 16.2 16.85 21.3 19.56 19.6 18.3

Correlation Coefficient

0.881

y = 0.5205x + 7.883R2 = 0.7767

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Female size (mm)

Mal

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mm

) Regression

To see if two variables vary together

To see how one variable affect another.

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Any Questions?