Turning Point

• At the beginning of the course, we discussed three ways in which mathematics and statistics can be used to facilitate psychological science

– quantification and measurement

– theoretical modeling

– evaluating theoretical models

• Up to this point, we have focused on quantification and measurement

Turning Point

• For the second part of the course, we are going to focus on ways in which statistics can be used to model psychological processes

• We will begin with some simple models and work our way up to more complex ones– modeling the influence of one variable on an outcome

(simple regression)– modeling the influence of many variables on an

outcome (multiple regression)

Example

• Let’s say we wish to model the relationship between coffee consumption and happiness

Some Possible Functions

Lines

• Linear relationships

• Y = a + bX

– a = Y-intercept (the value of Y when X = 0)

– b = slope (the “rise over the run”, the steepness of the line); a weight

Y = 1 + 2X

Lines and intercepts

• Y = a + 2X

• Notice that the implied values of Y go up as we increase a.

• By changing a, we are changing the elevation of the line.

Y = 1 + 2X

Y = 3 + 2X

Y = 5 + 2X

Lines and slopes

• Slope as “rise over run”: how much of a change in Y is there given a 1 unit increase in X.

• As we move up 1 unit on X, we go up 2 units on Y

• 2/1 = 2 (the slope)

Y = 1 + 2X

run

rise

move from 0 to 1

rise from 1 to 3 (a 2 unit change)

Lines and slopes

• Notice that as we increase the slope, b, we increase the steepness of the line

COFFEE

HAPPINESS

-4 -2 0 2 4

-50

510

Y = 1 + 2X

Y = 1 + 4X

COFFEE

HAPPINESS

-4 -2 0 2 4

-50

510

Lines and slopes

• We can also have negative slopes and slopes of zero.

• When the slope is zero, the predicted values of Y are equal to a.

Y = a + 0X = a

b=2

b=4

b=0

b=-2

b=-4

Other functions

• Quadratic function

• Y = a + bX2

– a still represents the intercept (value of Y when X = 0)

– b still represents a weight, and influences the magnitude of the squaring function

Quadratic and intercepts

• As we increase a, the elevation of the curve increases

COFFEE

HAPPINESS

-4 -2 0 2 4

05

1015

2025

30

Y = 0 + 1X2

Y = 5 + 1X2

Quadratic and Weight

• When we increase the weight, b, the quadratic effect is accentuated

COFFEE

HAPPINESS

-4 -2 0 2 4

020

4060

80100

120

Y = 0 + 1X2

Y = 0 + 5X2

COFFEE

HAPPINESS

-4 -2 0 2 4

-100

-50

050

100

Quadratic and Weight

• As before, we can have negative weights for quadratic functions.

• In this case, negative values of b flip the curve upside-down.

• As before, when b = 0, the value of Y = a for all values of X.Y = 0 – 5X2

Y = 0 – 1X2

Y = 0 + 1X2

Y = 0 + 5X2

Y = 0 + 0X2

Linear & Quadratic Combinations

• When linear and quadratic terms are present in the same equation, one can derive j-shaped curves

• Y = a + b1X + b2X2

linear weight (b)

- 0 +

quad

rati

c w

eigh

t (c)

+

0

-

Y = 5e-.8

• Exponential function

• Y = ae-bx

• In this equation e is a constant, roughly equal to 2.718; it is the base of the natural logs

– a represents the intercept (value of Y when X = 0)

– b still represents a weight, and affects the rate of change in the curve

Exponential Function

x

y

0 2 4 6 8 10

01

23

45

x

y

0 2 4 6 8 10

01

23

45

Y = 5e-.8

Y = 3e-.8

Y = 1e-.8

Y = 5e-.02

Y = 5e-.8

Y = 5e-5

Different values of a lead to different starting points

Different values of b lead to different rates of change

Exponential Function

Y = 5e .21

• Y = ae-bx

• Positive values of b lead to growth rather than decay

• One may use positive values of b to model growth processes (e.g., amount of novel words spoken by a child)

• One may use negative values of b to model decay processes (e.g., the forgetting of studied material over time)

Exponential Function

• When b is positive, the exponential function will grow without bounds

• Sometimes we want to limit the growth.

• One way to do so is to work with an alternative model, sometimes called the sigmoid function or the S-curve

XbaeY

1

1

S-Curve

Different values of a lead to rates of change

Different values of b lead to different locations on the X-axis

a = large

a = small

b = -5

b = +5

S-Curve

Some terminology

• When the relation between variables are expressed in this manner, we call the relevant equation(s) mathematical models

• The intercept and weight values are called parameters of the model.

• Although one can describe the relationship between two variables in the way we have done here, for now on we’ll assume that our models are causal models, such that the variable on the left-hand side of the equation is assumed to be caused by the variable(s) on the right-hand side.

Terminology

• The values of Y in these models are often called predicted values, sometimes abbreviated as Y-hat or . Why? They are the values of Y that are implied by the specific parameters of the model.

Y

Estimation

• Up to this point, we have assumed that our models are correct.

• There are two important issues we need to deal with, however:

– Is the gist of the model correct? That is, is a linear, as opposed to a quadratic, model the appropriate model for characterizing the relationship between variables?

– Assuming the model is correct, what are the correct parameters for the model?

Estimation

• For the next few weeks we will assume that the basic model (i.e., whether it is linear, whether the right variables are included) is correct. In the third part of the course, we will deal with methods for addressing this issue and comparing alternative models.

• The process of obtaining the correct parameter values (assuming we are working with the right model) is called parameter estimation.