161'130 Biometrics Week 3 Lecture slides

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161.130 Biometrics Week 3 Lecture slides

• Exploring Bivariate Data
• Scatterplots
• Text section 4.1
• CAST section 4.1
• Least Squares Nonlinear relationships
• Text section 4.3 and 4.4.2
• CAST sections 4.3 and 4.4
• Correlation
• Text section 4.2 and 4.4.1
• CAST section 4.2 and 4.5
• Multivariate Data
• CAST section 4.6

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• Univariate Data
• Single measurement from each individual
• Cannot associate variation in that measurement
  with other characteristics of the individuals
• Bivariate Data
• Two measurements from each individual
• May be be able to associate variation in one
  measurement with changes in the other measurement
• Examples are ...
• Blood pressure and weight of males in their 50s
• Carbohydrate content and moisture content of corn
  
• Our aim with such data is to find information
  about the relationship between the variables.

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Three Tools we will use

• Scatterplot, a two-dimensional graph of data
  values
• Correlation, a statistic that measures the
  strength and direction of a linear relationship
• Regression Equation, an equation that describes
  the average relationship between a response and
  explanatory variable

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Scatterplots

• The relationship between two variables cannot be
  determined from examination of the two variables
  in isolation.

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Scatterplots
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Scatterplots
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Scatterplots
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Scatterplots
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Scatterplots
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Looking for Patterns in Scatterplots

• What is the average pattern? Does it look like a
  straight line or is it curved?
• What is the direction of the pattern?
• How much do individual points vary from the
  average pattern?
• Are there any unusual data points?

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Positive/Negative Association

• Two variables have a positive association when
  the values of one variable tend to increase as
  the values of the other variable increase.
• Two variables have a negative association when
  the values of one variable tend to decrease as
  the values of the other variable increase.

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Example Height and Handspan

• Data shown are the first 12 observations of a
  data set that includes the heights (in inches)
  and fully stretched handspans (in centimeters) of
  167 college students.

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Example Height and Handspan

• Taller people tend to have greater handspan
  measurements than shorter people do.
• When two variables tend to increase together, we
  say that they have a positive association.
• The height and handspan measurements appear to
  have a linear relationship.

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Example Driver Age and MaximumReading
Distance of Highway Signs

• A research firm determined the maximum
  distance at which each of 30 randomly selected
  drivers could read a newly designed road sign.
• The 30 participants in the study ranged in
  age from 18 to 82 years old.
• We want to examine the relationship between
  age and the sign legibility (readability)
  distance.

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Example Driver Age and MaximumReading
Distance of Highway Signs

• We see a negative association with a linear
  pattern.
• We will use a straight-line equation to model
  this relationship.

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Identifying Groups and Outliers

• Use different plotting symbols or colours to
  represent different subgroups.
• Look for outliers points that have an unusual
  combination of data values.

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• In both univariate and bivariate data sets,
  outliers or clusters must be very distinct before
  we should conclude that they are real, in the
  absence of further external information
  confirming that the individuals are distinct.
• Particularly in small data sets, outliers,
  clusters and other patterns may arise by chance,
  without being associated with any real features
  in the individuals.
• Be careful not to over-interpret features in
  scatterplot unless they are well defined,
  especially if the sample size is small.

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Describing Linear Patterns using a Regression
Line
When the best equation for describing the
relationship between x and y is a straight line,
the equation is called the regression line.

• Two purposes of a regression line
• to estimate the average value of y at any
  specified value of x
• to predict the value of y for an individual,
  given that individuals x value

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Example Body Mass Index (BMI)

• Body Mass Index is calculated from the
  correlation between Height and body mass
  (weight)

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Example Height and Handspan

• Regression equation Handspan -3.0 0.35
  Height
• Estimate the average handspan for people 60
  inches tall (1.50m)

Average handspan -3.0 0.35(60) 18 cm.
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Example Height and Handspan (cont)

• Regression equation Handspan (cm) -3 0.35
  Height

Slope 0.35 Handspan increases by 0.35 cm, on
average, for each increase of 1 inch in height.
Intercept - 3.0
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Examples for you to work out
Knowing how to set up bivariate data
A table of bivariate data given are Draw a
scatterplot, using these data.
Solutions
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The Equation for the Regression Line

• is spoken as y-hat, and it is also referred
  to either as predicted y or estimated y.
• b 0 is the intercept of the straight line. The
  intercept is the value of y when x 0.
• b1 is the slope of the straight line. The slope
  tells us how much of an increase (or decrease)
  there is for the y variable when the x variable
  increases by one unit. The sign of the slope
  tells us whether y increases or decreases when x
  increases.
• e random errors (residuals) with mean zero

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Example Driver Age and Maximum Legibility
Distance of Highway Signs

• Regression equation Distance 577 - 3 Age

Slope of 3 tells us that, on average, the
legibility distance decreases, on average, by 3
feet when age increases by one year
Estimate the average distance for 20-year-old
driversAverage distance 577 3(20) 517
ft. Predict the legibility distance for a
20-year-old driverPredicted distance 577
3(20) 517 ft.
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Extrapolation

• Usually a bad idea to use a regression equation
  to predict values far outside the range where the
  original data fell.
• No guarantee that the relationship will continue
  beyond the range for which we have observed data.

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Least Squares Line and Formulas

• Least Squares Regression Line minimizes the sum
  of squared prediction errors.
• SSE Sum of squared prediction errors.
• Formulas for Slope and Intercept

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Linear model

• Only appropriate when the cloud of crosses in a
  scatterplot of the data is regularly spread
  around a straight line.
• If the crosses are scattered round a curve, the
  relationship is called nonlinear and other models
  must be used.
• Outliers should be investigated
• Detecting problems with the model
• Plot residuals against X to look for problems in
  the model

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Nonlinear Relationships

• If the relationship between Y and X is nonlinear,
  a linear model will give poor predictions and
  must be avoided.
• Transformation of one or both variables
• often possible to linearise the relationship and
  therefore use least squares to fit a linear model
  to the transformed variables
• For many data sets, a logarithmic transformation
  works, but a more general power transformation is
  sometimes needed to linearise the relationship

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Example The Development of
Musical Preferences

• We want to examine the relationship between
  song-specific age (age in the year a particular
  song was popular) and musical preference
  (positive score gt above average, negative score
  gt below average).
• The 108 participants in the study ranged in age
  from 16 to 86 years old.

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Example The Development of
Musical Preferences

• Popular music preferences acquired in late
  adolescence and early adulthood.
• The association is nonlinear.

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Example The Development of
Musical Preferences

• Interpretation If you are 10 now and a song came
  out in 1967 you were -30 years old when the song
  came out.
• If you are 85 now and a song comes out, you are
  85.
• So the 85 year olds don't like current music, the
  young ones don't like old music etc.

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Measuring Strength and Direction with the
Correlation Coefficient
Correlation Coefficient r indicates the strength
and the direction of a straight-line relationship.

• The strength of the relationship is determined by
  the closeness of the points to a straight line.
• The direction is determined by whether one
  variable generally increases or generally
  decreases when the other variable increases.

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Interpretation of r and a Formula

• r is always between 1 and 1
• magnitude indicates the strength
• r 1 or 1 indicates a perfect linear
  relationship
• sign indicates the direction
• r 0 indicates a slope of 0 so a change of x
  does not change the predicted value of y
• Formula for Pearson Linear Correlation
  Coefficient

r
.
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Interpretation of r and its Formula

• Formula for Pearson Linear Correlation
  Coefficient r
• PS You will not be required to use this formula,
  but in an exam the summary numbers and
  formula may be given to you so you can check your
  answer.

r
.
PS You will not be required to use this formula,
but in an exam the summary numbers and
formula may be given to you so you can check your
answer.
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Interpretation of r
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Interpretation of r (contd)
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Interpretation of r (contd)

• Pearsons Linear Correlation Coefficient
• The correlation r is a measure of how close Y
  and X are to having a straight-line association.
  
• r 1 implies a perfect straight-line
  relationship with positive slope
• r 0.9 implies a very strong straight-line
  relationship with positive slope
• r gt 0.7 implies a moderate straight-line
  relationship with positive slope
• 0.3 lt r 0.7 implies a weak to moderate
  straight-line relationship with positive slope
• 0lt r 0.3 implies a very weak straight-line
  relationship with positive slope
• r 0 implies no linear relationship or zero
  slope.
• Ditto r lt 0, but implying relationship with
  negative slope

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Interpretation of r (contd)

• Pearsons Linear Correlation Coefficient r
• The correlation r is a measure of how close Y
  and X are to having a straight-line association.
  In summary

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Example Height and Handspan (cont)

• Regression equation Handspan -3 0.35 Height
  
• Correlation r 0.74
• a medium/strong positive linear
  relationship.

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Example Driver Age and Maximum Legibility
Distance of Highway Signs (cont)

• Regression equation Distance 577 - 3 Age

Correlation r - 0.8 gt a strong
negative linear association.
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Example Left and Right Handspans

• If you know the span of a persons right hand,
  can you accurately predict his/her left handspan?

Correlation r 0.95 gt a very strong
positive linear relationship.
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Example Verbal SAT and GPA

• Grade point averages (GPAs) and verbal SAT
  (Standardised college Admission Test) scores for
  a sample of 100 university students.
• Correlation r 0.485 gt
• a moderately strong positive linear
  relationship.

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Example Age and Hours of TV Viewing

• Relationship between age and hours of daily
  television viewing for 1913 survey respondents.
• Correlation r 0.12 a weak connection.
• Note some participants claimed to watch more
  than 20 hours/day!

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Example Hours of Sleep and Hours of Study

• Relationship between reported hours of sleep the
  previous 24 hours and the reported hours of
  study during the same period for a sample of 116
  college students.

Correlation r 0.36 a weak to moderate
negative association.
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Example Driver Age and Maximum Legibility
Distance of Highway Signs (cont)

• Check the Regression Model by Calculating
  Residuals

So we can compute the residuals for all 30
observations. Positive mean residual gt observed
values higher than predicted. Negative mean
residual gt observed values lower than predicted.
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Why the Models may not always make Sense

• Allowing outliers to overly influence the
  results
• Combining groups inappropriately
• Using correlation and a straight-line equation to
  describe curvilinear data

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Example Height and Foot Length (cont)
Three outliers were data entry errors.

• Regression equation uncorrected data 15.4
  0.13 height corrected data -3.2 0.42 height
  
• Correlation uncorrected data r
  0.28 corrected data r 0.69

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Example Earthquakes in US
San Francisco earthquake of 1906.

• Correlation coefficient
• Inclusive of all data r 0.73
• Without San Francisco r 0.96

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Example Height and Heavy Feet

• Scatterplot of all data College student heights
  and responses to the question What is the
  fastest you have ever driven a car?

Scatterplot by gender Combining two groups led
to illegitimate correlation
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Example Dont Predict without drawing a Plot
Population of US (in millions) for each census
year between 1790 and 1990.

• Correlation r 0.96Regression Line
  population 2218 1.218(Year)Poor Prediction
  for Year 2005 2218 1.218(2005), about 224
  million, which is less than the 1990 population.