Transcript Parametric Inference

Parametric Inferential
Statistics
Types of Inference
• Estimation: On the basis of information in
a sample of scores, we estimate the value
of a population parameter.
• Hypothesis Testing: We determine how
well the data in the sample fit with the
hypothesis that in the population a
parameter has a particular value.
Sampling Distribution
• The probability distribution of a statistic.
• Imagine this:
– Draw an uncountably large number of
samples, each with N scores.
– For each sample, calculate a statistic (for
example, the mean).
– The distribution of these statistics is the
sampling distribution
Desirable Properties of Estimators
• CURSE
– Consistency
– Unbiasedness
– Resistance
– Sufficiency
– Efficiency
Unbiasedness
• Unbiasedness: The expected value
(mean) of the statistic is exactly equal to
the value of the parameter being
estimated.
• The sample mean is an unbiased
estimator of the population mean.
• The sample variance is an unbiased
estimator of the population variance
• The sample standard deviation is not an
absolutely unbiased estimator of the
population standard deviation.
• Consider this sampling distribution of
variances:
Sample Variance
Probability
2
.5
4
.5
Population Variance =
E(s2)=ƩPis2
1
2
3
The population standard deviation must be
SQRT(3) = 1.732.
• For these same samples, the sampling
distribution of the standard deviations is:
Sample SD
Probability
SQRT(2) = 1.414
.5
SQRT(4) = 2
.5
Population Variance =
E(s)=ƩPis
.707
1
1.707
• Oops, the expected value (1.707) is not
equal to the value of the population
parameter (1.732).
Relative Efficiency
• The standard deviation of a sampling
distribution is called its standard error.
• The smaller the standard error, the less
error one is likely to make when using the
statistic to estimate a parameter.
• Statistics with relatively low standard
errors are known as efficient statistics.
We Play a Little Game
• You are allowed to drawn a sample of
scores from a population whose mean
Professor Karl knows.
• You then get to estimate that mean from
the sample data.
• If your estimate is within one point of the
true value of the parameter, you get an A
in the course; otherwise you fail.
• There are three different estimators
available, X, Y, and Z.
• Each is absolutely unbiased.
• They differ greatly in dispersion.
• Which one will you use?
• Let’s look at their sampling distributions
Estimator X: SEM = 5
You have a 16% chance of getting that A.
Estimator Y: SEM = 1
You have a 68% chance of earning that A.
Estimator Z: SEM = 0.5
You have a 95% chance of earning that A.
Consistency
• With a consistent estimator, the standard
error decreases as the sample size
increases.
• For example, for the standard error of the
mean:
M 

n
Sufficiency
• A sufficient estimator uses all of the
information in the sample.
• Consider the range. It uses information
from the lowest score and the highest
score.
• Consider the variance. It uses information
from every score.
Resistance
• This is with respect to resistance to the
effects of outliers.
• We have already seen that the median is
much more resistant than is the mean.
Parameter Estimation
• There are two basic types
– Point Estimation: We come up with a single
value which is our best bet regarding what the
value of the parameter is.
– Interval Estimation: We come up with an
interval of values which we are confident
contains the true value of the parameter.
Confidence Coefficient
• CC is the subjective probability that the
interval will include the true value of the
parameter.
• Suppose CC = .95. Were we to construct
an infinite number of confidence intervals,
95% of them would include the true value.
• α is the subjective probability that the
interval will not contain the true value.
If Sampling Distribution is Normal
• The confidence interval will be
ˆ  z ˆ  ˆ  z ˆ
• Where theta-hat is the estimator
• Sigma theta-hat is the standard error
• And z is the how far you need go out from
the mean of the sampling distribution to
encompass the middle CC proportion of
the distribution.
Example
• At Bozo College, IQ is normally distributed
with a standard deviation of 15.
• We want to estimate the mean.
• A sample of 1 has mean = 110.
• A 95% CI for the mean is 110  1.96(15) =
[80.6, 139.4].
Hypothesis Testing
• The null hypothesis states that some
parameter has a particular value.
• Example:  = 100
• The alternative hypothesis states that
the null hypothesis is not correct.
• Example:   100
• These are nondirectional hypotheses –
the alternative does not predict the
direction of difference from 100.
Directional Hypotheses
• The alternative does predict a direction.
• Example
– H is   100
– H1 is  > 100
• Another example
– H is  100
– H1 is  < 100
What the Null Usually Is
• It is usually that the value of the correlation
between two variables or sets of variables
is zero.
• That is, the variables are not related.
• The researcher usually thinks that the null
is incorrect.
What the Null Rarely Is
• The prediction from a mathematical model
being tested.
• For example, mean weight loss during this
treatment is 23.67 pounds.
• In this case, if we find the null to be
substantially incorrect, we need to reject or
revise the mathematical model (aka
“theory”).
Testing the Null
• Gather relevant data
• Determine how well the data fit with the
null hypothesis, using a statistic called “p,”
the level of significance.
• p is the probability of obtaining a
sample as or more discrepant with the
H than is that which we did obtain,
assuming that the H is true.
• The higher this p, the better the fit
between the data and the H.
• If this p is low we have cast doubt upon
the H.
• If p is very low, we reject the H. How low
is very low?
• Very low is usually .05 -- the decision rule
most often used in Psychology is Reject
the null if p  .05.
Simple Example
•
•
•
•
•
•
H: mean IQ of my extended family is 145.
H1: no it isn’t
Sample of one score: IQ = 110.
Assume SD = 15 and population normal.
z = (110 - 145) / 15 = -2.33
Is that z score unusual enough for us to
reject the null hypothesis?
• From the normal curve table, we
determine that were the null true, we
would get a z that far from zero only
1.98% of the time.
• That is, p = .0198.
• This is less than .05, so we reject the null
hypothesis and conclude that the mean is
not 145.
This Was a Two-Tailed Test
Do You Need a p Value?
• Journals act like it is not real science
unless there is a p value,
• but all you really need is a CI.
• 95% CI = 110  1.96(15) = [80.6, 139.4]
• We are 95% confident that the mean is
between 80.6 and 139.4.
• That excludes 145, so we at least 95%
confident that the mean is not 145 
Reject the null.
A More Traditional Approach
• Think about the sampling distribution of
the test statistic (z).
• The nonrejection region is the area with
values that would not lead to rejection of
the null.
• The rejection region is the area with
values that would lead to rejection of the
null.
CV is the “critical value,” the boundary
between rejection and nonrejection regions
Decision Rule
• If |z| > 1.96, then p < .05 and we reject the
null.
• Otherwise, we retain the null.
• We never figure out exactly what the value
of p is.
• I strongly prefer that you use the modern
approach, where you find the exact value
of p.
“Confusion Matrix”
Decision
Reject H
The True Hypothesis Is
The H1
The H
correct decision Type I error
Assert H1
(power)
Retain H
Type II error
Do not assert H1
()
()
correct
decision
(1- )
Signal Detection
Prediction
Signal is
there
Signal is
not there
Is the Signal Really There?
Yes
No
True Positive
False Positive
[hit]
()
(power)
False Negative
True Negative
[miss]
(1- )
()
Relative Seriousness of Type I and II Errors
• Tumor rate in rats is 10%.
• Treat them with suspect drug.
– H: rate  10%; drug is safe
– H1: rate > 10%; drug is not safe
• Type I Error: The drug is safe, but you
conclude it is not.
• Type II Error: The drug is not safe, but
you conclude it is.
• Testing experimental blood pressure drug.
– H: Drop in BP  0; drug does not work
– H1: Drop in BP > 0; drug does work
• Type I Error: The drug does not lower BP,
but you conclude it does.
• Type II Error: The drug does lower BP,
but you conclude it does not.
Directional Hypotheses
• The alternative hypothesis predicts the
direction of the difference between the
actual value of the parameter and the
hypothesized value.
• The rejection region will be in only one tail
– the side to which the alternative
hypothesis points.
• H :   145 versus H1:  > 145
• z = (110 - 145) / 15 = -2.33
• Our test statistic is not in the rejection
region, we must retain the null.
• The p value will be P(z > -2.33), which is
equal to .9901.
• The data fit very well with the null
hypothesis that   145 .
Change the Prediction
• H :   145 versus H1:  < 145.
• The rejection region is now in the lower
tail.
• If z  -1.645, we reject the null.
• Our z is still -2.33, we reject the null.
• The p value is now P(z < -2.33), which is
equal to .0099.
• We do not double the p value as we would
with nondirectional hypotheses.
Pre- or Post-diction?
• If you correctly predicted (H1) the direction,
the p value will be half what it would have
been with nondirectional hypotheses.
• That gives you more power.
• BUT others will suspect that you
postdicted the direction of the difference.
Frequency of Type I Errors
• If we are in that universe where the null
hypothesis is always true, using the .05
criterion of statistical significance,
• We should make Type I errors 5% of the
time.
• There may be factors that inflate this
percentage
• Failures to reject the null are not often
published.
• “Publish or Perish.”
• May produce unconscious bias against
keeping the null, affecting data collection
and analysis.
• May lead to fraud.
The File Drawer Problem
• 100 researchers test the same absolutely
true null hypothesis.
• 5 get “significant” results. Joyfully, they
publish, unaware that their conclusions are
Type I errors.
• The other 95 just give up and put their “not
significant” results in a folder in the filing
cabinet.
Is an Exact Null Ever True?
• The null is usually that the correlation
between two things is zero.
• Correlation coefficients are continuously
distributed between -1 and +1.
• The probability of any exact value of a
continuous variable (such as  = 0) is
vanishingly small.
• But a null may be so close to true that it
might as well be true.
Doggie Dance