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```Correlation and Regression
Correlation and Regression
• Used when we are interested in the relationship
between two variables.
• NOT the differences between means or medians of
different groups.
Correlation and Regression
• Used when we are interested in the relationship
between two variables.
• NOT the differences between means or medians of
different groups.
The reverse is also true… so in your paper, you should not have written:
“There was a correlation between number of pupae and presence of an
interspecific competitor.”
Rather, the correct way would be:
“There was a difference between the mean number of pupae produced
between treatments with and without an interspecific competitor.”
Correlation
• This is used to:
- describe the strength of a relationship
between two variables…. This is the “r value” and it
can vary from -1.0 to 1.0
Correlation
• This is used to:
- describe the strength of a relationship
between two variables…. This is the “r value” and it
can vary from -1.0 to 1.0
- determine the probability that two
UNRELATED variables would produce a relationship
this strong, just by chance. This is the “p value”.
IF N = 62, then rcrit = 0.250 for p = 0.05, rcrit = 0.325 for p = 0.01
Correlation
• Important Note:
– Correlation does not imply causation - the variables are
related, but one does not cause the second.
“spurious” correlation
Correlation
• Important Note:
– Correlation does not imply causation - the variables are
related, but one does not cause the second.
– Often, the variables are both dependent variables in the
experiment… such as mean mass of flies and number of
offspring.
- so it is incorrect to think of one variable as ‘causing’ the
other….
As number increases, amount of food per individual declines, and
flies grow to a smaller size.
Or, as flies grow, small ones need less and so more small ones
can survive together than large ones.
Correlation
• Parametric test - the Pearson Correlation
coefficient.
– If the data is normally distributed, then you can
use a parametric test to determine the
correlation coefficient - the Pearson correlation
coefficient.
negative
NOTE: no lines drawn through points!
Pearson’s Correlation
• Assumptions of the Test
– Random sample from the populations
– Both variables are approximately normally
distributed
– Measurement of both variables is on an interval
or ratio scale
– The relationship between the 2 variables, if it
exists, is linear.
• Thus, before doing any correlation, plot the
relationship to see if its linear!
Pearson’s Correlation
• How to calculate the Pearson’s correlation coefficient
r
x y

 xy 
n
2
2
(
x
)
(
y
)
( x 2  
)(  y 2  
)
n
n
n = sample size
Testing r
n2
tr
2
1 r
•
•
•
•
Calculate t using above formula
Compare to tabled t-value with n-2 df
Reject null if calculated value > table value
But SPSS will do all this for you, so you don’t need to!
Example
• The heights and arm spans of 10 adult males
were measured in cm. Is there a correlation
between these two measurements?
Example
Height (cm)
Arm Span (cm)
171
173
195
193
180
188
182
185
190
186
175
178
177
182
178
182
192
198
202
202
Step 1 – plot the data
205
arm span
200
195
190
185
180
175
170
165
170
175
180
185
Height
190
195
200
205
Example
• Step 2 – Calculate the correlation coefficient
- r = 0.932
• Step 3 – Test the significance of the relationship
- p = 0.0001
Nonparametric correlation
• Spearman’s test
• This is the most commonly used test when one of
the assumptions of the parametric test cannot be
met - usually because it is non-normal, non-linear, or
uses ordinal data.
• The only assumptions of the Spearman’s r test is that
the data is randomly collected and that the scale of
measurement is at least ordinal.
Spearman’s test
• Like most non-parametric tests, the data are first
ranked from smallest to largest
– in this case, each column is ranked independently of the
other.
• Then (1) subtract each rank from the other, (2)
square the difference, (3) sum the values, and (4)
plug into the following formula to calculate the
Spearman correlation coefficient.
Spearman’s test
• Calculating Spearman’s correlation coefficient
rs  1  (
6 d
2
n(n  1)
2
)
Testing r
• The null hypothesis for a Spearman’s
correlation test is also that:
–  = 0; i.e., H0:  = 0; HA:  ≠ 0
• When we reject the null hypothesis we can
accept the alternative hypothesis that there is
a correlation, or relationship, between the
two variables.
Testing r
n2
tr
2
1 r
•
•
•
•
Calculate t using above formula
Compare to tabled t-value with n-2 df
Reject null if calculated value > table value
But SPSS will do all this for you, so you don’t need to!
Example
• The mass (in grams) of 13 adult male tuataras
and the size of their territories (in square
meters) was measured. Are territory size and
the size of the adult male tuatara related?
Example
Observation number
Mass
Territory size
1
510
6.9
2
773
20.6
3
840
17.2
4
505
6.7
5
765
20
6
780
24.1
7
235
1.5
8
790
13.8
9
440
1.7
10
435
2.1
11
815
20.2
12
460
3.0
13
697
10.3
Step 1 – plot the data
30
territory size
25
20
15
10
5
0
0
200
400
600
mass
Note - not very linear
800
1000
number
Mass
mRANK
Territory
tRANK
d
d2
1
510
6
6.9
6
0
0
2
773
10
20.6
12
2
4
3
840
13
17.2
9
4
16
4
505
5
6.7
5
0
0
5
765
8
20
11
3
9
6
780
9
24.1
13
4
16
7
235
1
1.5
1
0
0
8
790
11
13.8
8
3
9
9
440
3
1.7
2
1
1
10
435
2
2.1
3
1
1
11
815
12
20.2
10
2
4
12
460
4
3.0
4
0
0
13
697
7
10.3
7
0
0
rs  1  (
6 d
2
n(n  1)
2
) =r
s
=1-
6(60)
13(168)
= 0.835
Example
• Step 2 – Calculate the correlation coefficient
• Step 3 – Test the significance of the relationship
 = 0.835, p = 0.001
n2
tr
2
1 r
= 5.03
Linear Regression
• Here we are testing a causal relationship
between the two variables.
• We are hypothesizing a functional
relationship between the two variables that
allows us to predict a value of the dependent
variable, y, corresponding to a given value of
the independent variable, x.
Regression
• Unlike correlation, regression does imply causality
• An independent and a dependent variable can be identified
in this situation.
– This is most often seen in experiments, where you experimentally
assign the independent variable, and measure the response as the
dependent variable.
• Thus, the independent variable is not normally distributed
(indeed, it has no variance associated with it!) - as it is usually
selected by the investigator.
Linear Regression
• For a linear regression, this can be written as:
–
–
–
–
y =  + x (or y = mx + b)
where y = population mean value of y at any value of x
 = the population (y) intercept, and
 = population slope.
• You can use this equation to make predictions although of course these are usually estimated by
sample statistics rather than population parameters.
Linear Regression
• Assumptions
– 1. The independent variable (X) is fixed and
measured without error – no variance.
– 2. For any value of the independent variable (X),
the dependent variable (Y) is normally
distributed, and the population mean of these
values of y, y is:
• y =  + x
Linear Regression
• Assumptions
– 3. For any value of x, any particular value of y is:
• yi =  + x + e
• Where e, the residual, is the amount by which
any observed value of y differs from the mean
value of y (analogous to “random error”)
• Residuals will follow a standard normal
distribution
Linear Regression
• Assumptions
– 4. The variances of the y variable for all
values of x are equal
– 5. Observations are independent – each
individual is measured only once.
OK
Y
X
Not OK
Y
X
Estimating the Regression Function
and Line
• A regression line always passes through the
point: “mean x, mean y”.
Example - Juniper pythons
measured single, randomly selected snakes at different temperatures (one snake per temp).
Temperature (˚C)
Heart Rate
2
5
4
11
6
11
8
14
10
22
12
23
14
32
16
29
18
32
Mean (x) = 10
Mean (y) = 19.88
Example
35
Heart rate
30
25
20
15
10
5
0
0
5
10
Temperature
15
20
Example
Mean x = 10; Mean y = 19.88
35
Heart rate
30
25
20
15
10
How much each value of y (yi)
deviates from the mean of y… y – yi
5
0
0
5
10
15
Temperature
• The horizontal line represents a regression line for y when x
(temperature) is not considered.
• Residuals are very large!
20
Estimating the Regression Function
and Line
• To measure total error, you want to sum the residuals… but
they will cancel out… so you must square the differences,
then sum.
• Now we have the TOTAL SUM OF SQUARES (SST)
• The sum of squares of the residuals is thus:
SSYT   ( y  y) 2
• Thus, you see a lot of variance in y when x is
not taken into account. How much of the
variance in y can be attributed to the
relationship with x?
Example
Heart rate
Mean x = 10; Mean y = 19.88
40
35
30
25
20
15
10
5
0
The “line of best fit” minimizes the
residual sum of squares.
0
5
10
15
20
Temperature
The best fit line represents a regression line for y when x
(temperature) is considered.
Now the residuals are very small – in fact, the smallest sum possible.
Estimating the Regression Function
and Line
• This “line of best fit” minimizes the y sum of
squares, and accounts for how x, the independent
variable, influences y, the dependent variable.
• The difference between the observed values and
this “line of best fit” are the residuals – the “error”
left over when the relationship is included.
Estimating the Regression Function
and Line
• The sum of squares of these regression residuals is
now:

SSY   ( y  y)
2
• This is equivalent to the ERROR SS = (SSe); it is the
variance “left over” after the realtionship with x has
been included.
Estimating the Regression Function
and Line
• How do we get this best fit line?
• Based on the principles we just went over,
you can calculate the slope and the intercept
of the best fit line.
Estimating the Regression Function
and Line
slope (b) 
x y

 xy 
n
2
( x )
2
x  n
intercept (a)  y  b x
Testing the Significance of the
Regression Line
• In a regression, you test the null hypothesis
– Hq:  = 0; HA:  ≠ 0
• This is done using an ANOVA procedure.
• To do this, you calculate sums of squares, their
corresponding degrees of freedom, mean squares,
and finally an F value (just like an ANOVA!)
Sums of Squares
• SSt - this is the value for sums of squares for y when
x is not considered (the total sums of squares)
• SSe - this is the value for the sums of squares of the
residuals - in other words, it represents the variance
in y that is still present even when x is considered
(the error sums of squares)
• SSr – this is the variation in y accounted for by the
relationship with x. It can be calculated two ways:
- by subtraction (SSt – SSe)
- directly using formula
Sums of Squares
SST   y 
2
SS R
( y )
2
n
x y

 b( xy 
)
SS E  SST  SS R
n
Regression ANOVA Table (see p. 120)
Source of
Variation
Sum of
Squares
Regression SSR
df
1
MS
SSR
Error
SSE
n-2 SSE/n-2
Total
SST
n-1 SST/n-1
F
MSR/MSE
Testing the Significance of the
Regression Line
• Interpret exactly as for an ANOVA
Coefficient of determination
• The coefficient of determination, or r2, tells
you what proportion of the variance in y is
explained by its dependence on x.
• r2 = SSR/SST
• e.g., if r2 = 0.98, then 98% of the variance in y
is dependent on x - or 2% of the variance is
unexplained.
Example
• Suppose you want to describe the effects of temperature on
development time in Drosophila.
• You let flies lay eggs (on mushrooms in 30 vials) for one day
• You select 3 temperature treatments (20, 25, 30oC) and
randomly assign 10 vials to each treatment.
• You count the number of flies that emerge each day. From
these data, you compute two variables, number of flies and
mean number of days to develop.
• Number of flies is not a dependent variable, because this did
not vary as a consequence of temperature – eggs were laid
before vials were placed in the temperature treatments. But,
you know that the number of flies – and competitive stress –
might cause a change in developmental rate. So, it is a
potential correlate.
OUTPUT – Linear Regression
OUTPUT: Multiple Regression – Abundance and Temp
Multiple regression – Stepwise
Source
SS
df
Total
Abundance
Temp
Regression
Residual
274.855
152.535
95.048
247.855
27.271
29
1
1
2
27
MS
F
P
ANCOVA: Comparing means between treatments (NOT looking for
linear relationship), while accounting for variability due to correlated
variables.
ANOVA ALONE:
ANCOVA: Comparing means between treatments, while accounting
for variability due to correlated variables.
ANCOVA: Analysis of Covariance
Diffs in PUT mean male mass between treat 1 vs. 3
Diffs in PUT mean male mass between treat 1 vs. 3
```
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