### Statistical inference

Most of what we’ve covered in this book so far is about producing descriptive statistics: calculating means and medians, plotting data in various ways, and producing confidence intervals. The bulk of the rest of this book will cover statistical inference: using statistical tests to draw some conclusion about the data. We’ve already done this a little bit in earlier chapters by using confidence intervals to conclude if means are different or not among groups.

As Dr. Nic mentions in her article in the “References and further reading” section, this is the part where people sometimes get stumped. It is natural for most of us to use summary statistics or plots, but jumping to statistical inference needs a little change in perspective. The idea of using some statistical test to answer a question isn’t a difficult concept, but some of the following discussion gets a little theoretical. The video from the Statistics Learning Center in the “References and further reading” section does a good job of explaining the basis of statistical inference.

The most important thing to gain from this chapter
is an understanding of how to use the *p*-value, *alpha*,
and decision rule to test the null hypothesis. But once you are
comfortable with that, you will want to return to this chapter to have
a better understanding of the theory behind this process.

### Hypothesis testing

#### The null and alternative hypotheses

The statistical tests in this book rely on testing a null hypothesis, which has a specific formulation for each test. The null hypothesis always describes the case where e.g. two groups are not different or there is no correlation between two variables, etc.

The alternative hypothesis is the contrary of the null hypothesis, and so describes the cases where there is a difference among groups or a correlation between two variables, etc.

Notice that the definitions of null hypothesis
and alternate hypothesis have nothing to do with what you want to find
or don't want to find, or what is interesting or not interesting, or
what you expect to find or what you don’t expect to find. If you
were comparing the height of men and women, the null hypothesis would
be that the height of men and the height of women were not different.
Yet, you might find it surprising if you found this hypothesis to be
true for some population you were studying. Likewise, if you were
studying the income of men and women, the null hypothesis would be that
the income of men and women are not different, in the population you
are studying. In this case you might be *hoping* the null
hypothesis is true, though you might be *unsurprised* if the alternative
hypothesis were true. In any case, the null hypothesis will take
the form that there is no difference between groups, there is no correlation
between two variables, or there is no effect of this variable in our
model.

#### p-value definition

Most of the tests in this book rely on using a
statistic called the *p*-value to evaluate if we should reject,
or fail to reject, the null hypothesis.

*Given the assumption that the null hypothesis
is true*, the *p*-value is defined as the probability of obtaining
a result equal to or more extreme than what was actually observed in
the data.

We’ll unpack this definition in a little bit.

##### Decision rule

The *p*-value for the given data will be determined
by conducting the statistical test.

This *p*-value is then compared to a pre-determined
value *alpha*. Most commonly, an *alpha* value of 0.05
is used, but there is nothing magic about this value.

If the *p*-value for the test is less than
*alpha*, we reject the null hypothesis.

If the *p*-value is greater than or equal
to *alpha*, we fail to reject the null hypothesis.

##### Coin flipping example

For an example of using the *p*-value for
hypothesis testing, imagine you have a coin you will toss 100 times.
The null hypothesis is that the coin is fair—that is, that it is equally
likely that the coin will land on heads as land on tails. The
alternative hypothesis is that the coin is not fair. Let’s say
for this experiment you throw the coin 100 times and it lands on heads
95 times out of those hundred. The *p*-value in this case
would be the probability of getting 95, 96, 97, 98, 99, or 100 heads,
or 0, 1, 2, 3, 4, or 5 heads, *assuming that the null hypothesis is
true*.

This is what we call a two-sided test, since we are testing both extremes suggested by our data: getting 95 or greater heads or getting 95 or greater tails. In most cases we will use two sided tests.

You can imagine that the *p*-value for this
data will be quite small. If the null hypothesis is true, and
the coin is fair, there would be a low probability of getting 95 or
more heads or 95 or more tails.

Using a binomial test, the *p*-value is <
0.0001.

(Actually, R reports it as < 2.2e-16, which
is shorthand for the number in scientific notation, 2.2 x 10^{-16},
which is 0.00000000000000022, with 15 zeros after the decimal point.)

Assuming an *alpha* of 0.05, since the
*p*-value is less than *alpha*, we reject the null hypothesis.
That is, we conclude that the coin is not fair.

binom.test(5, 100, 0.5)

Exact binomial test

number of successes = 5,
number of trials = 100, p-value < 2.2e-16

alternative hypothesis:
true probability of success is not equal to 0.5

##### Passing and failing example

As another example, imagine we are considering two classrooms, and we have counts of students who passed a certain exam. We want to know if one classroom had statistically more passes or failures than the other.

In our example each classroom will have 10 students. The data is arranged into a contingency table.

__Classroom__
__Passed__ __Failed__

A
8 2

B
3 7

We will use Fisher’s exact test to test if there
is an association between *Classroom* and the counts of passed
and failed students. The null hypothesis is that there is no association
between *Classroom* and *Passed/Failed*, based on the relative
counts in each cell of the contingency table.

Input =("

Classroom Passed Failed

A
8 2

B
3 7

")

Matrix = as.matrix(read.table(textConnection(Input),

header=TRUE,

row.names=1))

Matrix

Passed Failed

A
8 2

B
3 7

fisher.test(Matrix)

Fisher's Exact Test for Count Data

p-value
= 0.06978

The reported *p*-value is 0.070. If
we use an *alpha* of 0.05, then the *p*-value is greater than
*alpha*, so we fail to reject the null hypothesis. That is,
we did not have sufficient evidence to say that there is an association
between *Classroom* and *Passed/Failed*.

More extreme data in this case would be if the counts in the upper left or lower right (or both!) were greater.

__Classroom__
__Passed__ __Failed__

A
9 1

B
3 7__Classroom__
__Passed__ __Failed__

A
10 0

B 3
7

and so on, with Classroom B...

In most cases we would want to consider as "extreme"
not only the results when Classroom A has a high frequency of passing
students, but also results when Classroom B has a high frequency of
passing students. This is called a two-sided or two-tailed test.
If we were only concerned with one classroom having a high frequency
of passing students, relatively, we would instead perform a one-sided
test. The default for the *fisher.test* function is two-sided,
and usually you will want to use two-sided tests.

__Classroom__
__Passed__ __Failed__

A
2 8

B
7 3__Classroom__
__Passed__ __Failed__

A
1 9

B
7 3__Classroom__
__Passed__ __Failed__

A
0 10

B
7 3

and so on, with
Classroom B...

In both cases, "extreme" means there is a stronger
association between *Classroom* and *Passed/Failed*.

### Theory and practice of using *p*-values

#### Wait, does this make any sense?

Recall that the definition of the *p*-value
is:

*Given the assumption that the null hypothesis is
true*, the *p*-value is defined as the probability of obtaining
a result equal to or more extreme than what was actually observed in
the data.

The astute reader might be asking herself, “If I’m trying to determine if the null hypothesis is true or not, why would I start with the assumption that the null hypothesis is true? And why am I using a probability of getting certain data given that a hypothesis is true? Don’t I want to instead determine the probability of the hypothesis given my data?”

The answer is *yes*, we *would* like
a method to determine the likelihood of our hypothesis being true given
our data, but we use the *Null Hypothesis Significance Test* approach
since it is relatively straightforward, and has wide acceptance historically
and across fields.

In practice we do use the results of the statistical tests to reach conclusions about the null hypothesis.

Technically, the *p*-value says nothing about the alternative hypothesis.
But logically, if the null hypothesis is rejected, then its logical
complement, the alternative hypothesis, is supported. Practically,
this is how we handle significant *p*-values, though this practical
approach generates disapproval in some theoretical circles.

#### Statistics is like a jury?

Note the language used when testing the null hypothesis.
Based on the results of our statistical tests, we either *reject
*the null hypothesis, or *fail to reject* the null hypothesis.

This is somewhat similar to the approach of a jury in a trial. The jury either finds sufficient evidence to declare someone guilty, or fails to find sufficient evidence to declare someone guilty.

Failing to convict someone isn’t necessarily the same as declaring someone innocent. Likewise, if we fail to reject the null hypothesis, we shouldn’t assume that the null hypothesis is true. It may be that we didn’t have sufficient samples to get a result that would have allowed us to reject the null hypothesis, or maybe there are some other factors affecting the results that we didn’t account for. This is similar to an “innocent until proven guilty” stance.

#### Errors in inference

For the most part, the statistical tests we use
are based on probability, and our data could always be the result of
chance. Considering the coin flipping example above, if we did
flip a coin 100 times and came up with 95 heads, we would be compelled
to conclude that the coin was not fair. But 95 heads *could*
happen with a fair coin strictly by chance.

We can, therefore, make two kinds of errors in testing the null hypothesis:

• A *Type I error* occurs when the null
hypothesis really is true, but based on our decision rule we reject
the null hypothesis. In this case, our result is a *false*
*positive*; we think there is an effect (unfair coin, association
between variables, difference among groups) when really there isn’t.
The probability of making this kind error is *alpha*, the same
*alpha* we used in our decision rule.

• A *Type II error* occurs when the null
hypothesis is really false, but based on our decision rule we fail to
reject the null hypothesis. In this case, our result is a *false
negative*; we have failed to find an effect that really does exist.
The probability of making this kind of error is called *beta*.

The following table summarizes these errors.

Reality

_____________________________________Decision of Test__
__Null is true__ __Null
is false__

Reject null hypothesis
Type I error
Correctly

(prob. = alpha)
reject null

(prob. = 1 – beta)

Retain null hypothesis Correctly Type
II error

retain null
(prob. = beta)

(prob. = 1 – alpha)

##### Statistical power

The statistical power of a test is a measure of
the ability of the test to detect a real effect. It is related
to the effect size, the sample size, and our chosen *alpha* level.

The effect size is a measure of how unfair a coin
is, how strong the association between two variables, or how large the
difference among groups. As the effect size increases or as the
number of observations we collect increases, or as the *alpha*
level decreases, the power of the test increases.

Statistical power in the table above is indicated
by *1 – beta*, which is the probability of correctly rejecting
the null hypothesis.

An example should make these relationship clear.
Imagine we are sampling a class of 7^{th} grade students for
their height. In reality, for this class, the girls are taller
than the boys, but the difference is small (that is the effect size
is small), and there is a lot of variability in students’ heights.
You can imagine that in order to detect the difference between girls
and boys that we would have to measure many students. If we fail
to sample enough students, we might make a Type II error. That
is, we might fail to detect the actual difference in heights between
sexes.

If we had a different experiment with a larger effect size—for example the weight difference between mature hamsters and mature hedgehogs—we might need fewer samples to detect the difference.

Note also, that our chosen *alpha* plays a
role in the power of our test, too. All things being equal, across
many tests, if we decrease our *alph*a, that is to insist on a
lower rate of Type I errors, we are more likely to commit a Type II
error, and so have a lower power. This is analogous to a case
of a meticulous jury that has a very high standard of proof to convict
someone. In this case, the likelihood of a false conviction is
low, but the likelihood of a letting a guilty person go free is relatively
high.

#### The 0.05 alpha value is not dogma

The level of *alpha* is traditionally set
at 0.05 in some disciplines, though there is sometimes reason to choose
a different value.

One situation in which the *alpha* level is
increased is in preliminary studies in which it is better to include
potentially significant effects even if there is not strong evidence
for keeping them. In this case, the researcher is accepting an
inflated chance of Type I errors in order to decrease the chance of
Type II errors.

Imagine an experiment in which you wanted to see
if various environmental treatments would improve student learning.
In a preliminary study, you might have many treatments, with few observations
each, and you want to retain any potentially successful treatments for
future study. For example, you might try playing classical music,
improved lighting, complimenting students, and so on, and see if there
is any effect on student learning. You might relax your *alpha*
value to 0.10 or 0.15 in the preliminary study to see what treatments
to include in future studies.

On the other hand, in situations where a Type I,
false positive, error might be costly in terms of money or people’s
health, a lower *alpha* can be used, perhaps, 0.01 or 0.001.
You can imagine a case in which there is an established treatment
for cancer, and a new treatment is being tested. Because the new
treatment is likely to be expensive and to hold people’s lives in the
balance, a researcher would want to be very sure that the new treatment
is more effective than the established treatment. In reality,
the researchers would not just lower the *alpha* level, but also
look at the effect size, submit the research for peer review, replicate
the study, be sure there were no problems with the design of the study
or the data collection, and weigh the practical implications.

##### The 0.05 alpha value is almost dogma

In theory, as a researcher, you would determine
the *alpha* level you feel is appropriate. That is, the probability
of making a Type I error when the null hypothesis is in fact true.

In reality, though, 0.05 is almost always used
in most fields for readers of this book. Choosing a different
*alpha* value will rarely go without question. It is best
to keep with the 0.05 level unless you have good justification for another
value, or are in a discipline where other values are routinely used.

##### Practical advice

One good practice is to report actual *p*-values
from analyses. It is fine to also simply say, e.g. “The dependent
variable was significantly correlated with variable *A* (*p
*< 0.05).” But I prefer when possible to say, “The dependent
variable was significantly correlated with variable *A* (*p*
= 0.026).

It is probably best to avoid using terms like “marginally
significant” or “borderline significant” for *p*-values less than
0.10 but greater than 0.05, though you might encounter similar phrases.
It is better to simply report the *p*-values of tests or effects
in straight-forward manner. If you had cause to include certain
model effects or results from other tests, they can be reported as e.g.,
“Variables correlated with the dependent variable with *p* <
0.15 were *A*, *B*, and *C*.”

### Philosophy of statistical analyses

#### Statistics is not like a trial

When analyzing data, the analyst should not approach the task as would a lawyer for the prosecution. That is, the analyst should not be searching for significant effects and tests, but should instead be like an independent investigator using lines of evidence to find out what is most likely to true given the data, graphical analysis, and statistical analysis available.

#### Practical significance and statistical significance

It is important to remember to not let *p*-values
be the only guide for drawing conclusions. It is equally important
to look at the size of the effects you are measuring, as well as take
into account practical considerations like the costs of choosing a certain
path of action.

For example, imagine we want to compare the SAT
scores of two SAT preparation classes with a *t*-test.

Class.A = c(1500, 1505, 1505, 1510, 1510, 1510, 1515,
1515, 1520, 1520)

Class.B = c(1510, 1515, 1515, 1520, 1520, 1520,
1525, 1525, 1530, 1530)

t.test(Class.A, Class.B)

Welch Two Sample t-test

t = -3.3968, df = 18,
p-value = 0.003214

mean of x mean of y

1511 1521

The *p*-value is reported as 0.003, so we
would consider there to be a significant difference between the two
classes (*p* < 0.05).

But we have to ask ourselves the practical question, is a difference of 10 points on the SAT large enough for us to care about? What if enrolling in one class costs significantly more than the other class? Is it worth the extra money for a difference of 10 points on average?

#### p-values and effect sizes

It should be remembered that *p*-values do
not indicate the size of the effect being studied. It shouldn’t
be assumed that a small *p*-value indicates a large difference
between groups, or vice-versa.

For example, in the SAT example above, the *p*-value
is fairly small, but the size of the effect (difference between classes)
in this case is relatively small (10 points, especially small relative
to the range of scores students receive on the SAT).

In converse, there could be a relatively large
size of the effects, but if there is a lot of variability in the data
or the sample size is not large enough, the *p*-value could be
relatively large.

In this example, the SAT scores differ by 100 points
between classes, but because the variability is greater than in the
previous example, the *p*-value is not significant.

Class.C = c(1000, 1100, 1200, 1250, 1300, 1300, 1400,
1400, 1450, 1500)

Class.D = c(1100, 1200, 1300, 1350, 1400, 1400,
1500, 1500, 1550, 1600)

t.test(Class.A, Class.B)

Welch Two Sample t-test

t = -1.4174, df = 18,
p-value = 0.1735

mean of x mean of y

1290 1390 )

boxplot(cbind(Class.C, Class.D))

The problem of multiple p-values

One concept that will be in important in the following
discussion is that when there are multiple tests producing multiple
*p*-values, that there is an inflation of the Type I error rate.
That is, there is a higher chance of making false-positive errors.

This simply follows mathematically from the definition
of *alpha*. If we allow a probability of 0.05, or 5% chance,
of making a Type I error for any one test, as we do more and more tests,
the chances that at least one of them having a false positive becomes
greater and greater.

*p*-value adjustment

One way we deal with the problem of multiple *
p*-values in statistical analyses is to adjust *p*-values
when we do a series of tests together (for example, if we are
comparing the means of multiple groups).

###### Don’t use Bonferroni adjustments

There are various *p*-value adjustments available
in R. In certain types of tests, in this book we use Tukey range
adjustment. In other cases, we will use FDR, which stands for
*false discovery rate*, and in R is an alias for the Benjamini
and Hochberg method.

Unfortunately, students in analysis of experiments
courses often learn to use Bonferroni adjustment for *p*-values.
This method is simple to do with hand calculations, but is excessively
conservative in most situations, and, in my opinion, antiquated.

There are other *p*-value adjustment methods,
and the choice of which one to use is dictated either by which are common
in your field of study, or by doing enough reading to understand which
are statistically most appropriate for your application.

#### Preplanned tests

The statistical tests covered in this book assume
that tests are preplanned for their *p*-values to be accurate.
That is, in theory, you set out an experiment, collect the data as planned,
and then say “I’m going to analyze it with kind of model and do these
post-hoc tests afterwards”, and report these results, and that’s all
you would do.

Some authors emphasize this idea of preplanned tests, but in reality, of course, researchers will often collect data, and then use an exploratory data analysis approach to see what the data suggests before heading into statistical analyses.

#### Exploratory data analysis

Exploratory data analysis often relies upon examining the data with plotting the data in various ways and using simple tests like correlation tests to suggest what statistical analysis makes sense.

Sometimes the preplanned approach is appropriate. If an experiment is set out in a specific design, then usually it is appropriate to use the analysis suggested by this design.

However, in general I recommend an exploratory approach. There may be interesting correlations hidden in the data. Or perhaps the distribution of the data didn’t allow for the originally planned analysis.

#### p-value hacking

It is important when approaching data from an exploratory
approach, to avoid committing *p*-value hacking. Imagine
the case in which the researcher collects many different measurements
across a range of subjects. The researcher might be tempted to
simply try different tests and models to relate one variable to another,
for all the variables. He might continue to do this until he found
a test with a significant *p*-value.

But this would be a form of *p*-value hacking.

Because an *alpha* value of 0.05 allows us
to make a false-positive error five percent of the time, finding one
*p*-value below 0.05 after several successive tests may simply
be due to chance.

Some forms of *p*-value hacking are more egregious.
For example, if one were to collect some data, run a test, and then
continue to collect data and run tests iteratively until a significant
*p*-value is found.

##### Publication bias

A related issue in science is that there is a bias
to publish, or to report, only significant results. This can also
lead to an inflation of the false-positive rate. As a hypothetical
example, imagine if there are currently 20 similar studies being conducted
testing a similar effect—let’s say the effect of glucosamine supplements
on joint pain. If 19 of those studies found no effect and so were
discarded, but one study found an effect using an *alpha* of 0.05,
and was published, is this really any support that glucosamine supplements
decrease joint pain?

### A few of xkcd comics

#### Significant

#### Null hypothesis

#### P-values

### Experiments, sampling, and causation

#### Types of experimental designs

##### Experimental designs

A true experimental design assigns treatments in a systematic manner. The experimenter must be able to manipulate the experimental treatments and assign them to subjects. Since treatments are randomly assigned to subjects, a causal inference can be made for significant results. That is, we can say that the variation in the dependent variable is caused by the variation in the independent variable.

For interval/ratio data, traditional experimental designs can be analyzed with specific parametric models, assuming other model assumptions are met. These traditional experimental designs include:

• Completely random design

• Randomized complete block design

• Factorial

• Split-plot

• Latin square

#### Quasi-experiment designs

Often a researcher cannot assign treatments to individual experimental units, but can assign treatments to groups. For example, if students are in a specific grade or class, it would not be practical to randomly assign students to grades or classes. But different classes could receive different treatments (such as different curricula). Causality can be inferred cautiously if treatments are randomly assigned and there is some understanding of the factors that affect the outcome.

#### Observational studies

In observational studies, the independent variables are not manipulated, and no treatments are assigned. Surveys are often like this, as are studies of natural systems without experimental manipulation. Statistical analysis can reveal the relationships among variables, but causality cannot be inferred. This is because there may be other unstudied variables that affect the measured variables in the study.

#### Sampling

Good sampling practices are critical for producing good data. In general, samples need to be collected in a random fashion so that bias is avoided.

In survey data, bias is often introduced by a self-selection bias. For example, internet or telephone surveys include only those who respond to these requests. Might there be some relevant difference in the variables of interest between those who respond to such requests and the general population being surveyed? Or bias could be introduced by the researcher selecting some subset of potential subjects, for example only surveying a 4-H program with particularly cooperative students and ignoring other clubs. This is sometimes called “convenience sampling”.

In election forecasting, good pollsters need to account for selection bias and other biases in the survey process. For example, if a survey is done by landline telephone, those being surveyed are more likely to be older than the general population of voters, and so likely to have a bias in their voting patterns.

#### Plan ahead and be consistent

It is sometimes necessary to change experimental conditions during the course of an experiment. Equipment might fail, or unusual weather may prevent making meaningful measurements.

But in general, it is much better to plan ahead and be consistent with measurements.

##### Consistency

People sometimes have the tendency to change measurement frequency or experimental treatments during the course of a study. This inevitably causes headaches in trying to analyze data, and makes writing up the results messy. Try to avoid this.

##### Controls and checks

If you are testing an experimental treatment, include
a *check* treatment that almost certainly will have an effect and
a *control* treatment that almost certainly won’t. A *control*
treatment will receive no treatment and a *check* treatment will
receive a treatment known to be successful. In an educational
setting, perhaps a control group receives no instruction on the topic
but on another topic, and the check group will receive standard instruction.

Including checks and controls helps with the analysis in a practical sense, since they serve as standard treatments against which to compare the experimental treatments. In the case where the experimental treatments have similar effects, controls and checks allow you say, for example, “Means for the all experimental treatments were similar, but were higher than the mean for control, and lower than the mean for check treatment.”

##### Include alternate measurements

It often happens that measuring equipment fails or that a certain measurement doesn’t produce the expected results. It is therefore helpful to include measurements of several variables that can capture the potential effects. Perhaps test scores of students won’t show an effect, but a self-assessment question on how much students learned will.

##### Include covariates

Including additional independent variables that might affect the dependent variable is often helpful in an analysis. In an educational setting, you might assess student age, grade, school, town, background level in the subject, or how well they are feeling that day.

The effects of covariates on the dependent variable may be of interest in itself. But also, including co-variates in an analysis can better model the data, sometimes making treatment effects more clear or making a model better meet model assumptions.

### Optional discussion: Alternative methods to the Null Hypothesis Significance Test

#### The NHST controversy

Particularly in the fields of psychology and education, there has been much criticism of the null hypothesis significance test approach. From my reading, the main complaints against NHST tend to be:

• Students and researchers don’t really understand
the meaning of *p*-values.

• *p*-values don’t include important information
like confidence intervals or parameter estimates.

• *p*-values have properties that may be
misleading, for example that they do not represent effect size, and
that they change with sample size.

• We often treat an *alpha *of 0.05 as
a magical cutoff value.

Personally, I don’t find these to be very convincing arguments against the NHST approach.

The first complaint is in
some sense pedantic: Like so many things, students and researchers
learn the definition of *p*-values at some point and then eventually
forget. This doesn’t seem to impact the usefulness of the approach.

The second point has weight only if researchers
use *only* *p*-values to draw conclusions from statistical
tests. As this book points out, one should always consider the
size of the effects and practical considerations of the effects, as
well present finding in table or graphical form, including confidence
intervals or measures of dispersion. There is no reason why parameter
estimates, goodness-of-fit statistics, and confidence intervals can’t
be included when a NHST approach is followed.

The properties in the third point also don’t count
much as criticism if one is using *p*-values correctly. One
should understand that it is possible to have a small effect size and
a small *p*-value, and vice-versa. This is not a problem,
because *p*-values and effect sizes are two different concepts.
We shouldn’t expect them to be the same. The fact that *p*-values
change with sample size is also in no way problematic to me. It
makes sense that when there is a small effect size or a lot of variability
in the data that we need many samples to conclude the effect is likely
to be real.

(One case where I think the considerations in the preceding point are commonly problematic is when people use statistical tests to check for the normality or homogeneity of data or model residuals. As sample size increases, these tests are better able to detect small deviations from normality or homoscedasticity. Too many people use them and think their model is inappropriate because the test can detect a small effect size, that is, a small deviation from normality or homoscedasticity).

The fourth point is a good one. It doesn’t
make much sense to come to one conclusion if our *p*-value is 0.049
and the opposite conclusion if our *p*-value is 0.051. But
I think this can be ameliorated by reporting the actual *p*-values
from analyses. And at some point, one does sometimes have to have
a cutoff value.

Overall it seems to me that these complaints condemn
poor practices that the authors observe: not reporting the size of effects
in some manner; not including confidence intervals or measures of dispersion;
basing conclusions solely on *p*-values; and not including important
results like parameter estimates and goodness-of-fit statistics.

#### Alternatives to the NHST approach

##### Estimates and confidence intervals

One approach to determining statistical significance is to use estimates and confidence intervals. Estimates could be statistics like means, medians, proportions, or other calculated statistics. To compare among group medians, one might calculate the medians for each group and a 95% confidence interval for each median. Then, medians with non-overlapping confidence intervals are considered statistically significantly different.

This approach is used in a few spots in this book,
including in the *Confidence Intervals* chapter and in the Jerry
Coyne example.

This approach can be very straightforward, easy for readers to understand, and easy to present clearly.

Another example of when this approach can be used
is in the case of estimates from an analysis like multiple regression. Instead
of reporting *p*-values for terms in the model, the estimates and
confidence intervals for each term in the model could be reported.
Terms with confidence intervals not including zero contribute statistically
to the model, and the value of slope coefficient suggests the size of
the contribution of the term.

##### Bayesian approach

The most popular competitor to the NHST approach
is Bayesian inference. Bayesian inference has the advantage of
calculating the probability of the hypothesis *given the data*,
which is what we thought we should be doing in the “Wait, does this
make any sense?” section above. Essentially it takes *prior*
knowledge about the distribution of the parameters of interest for a
population and adds the information from the measured data to reassess
some hypothesis related to the parameters of interest. If the
reader will excuse the vagueness of this description, it makes intuitive
sense. We start with what we suspect to be the case, and then
use new data to assess our hypothesis.

One disadvantage of the Bayesian approach is that it is not obvious in most cases what could be used for legitimate prior information. A second disadvantage is that conducting Bayesian analysis is not as straightforward as the tests presented in this book.

### References and further reading

** [Video] “Understanding statistical
inference”** from Statistics Learning Center (Dr. Nic). 2015.
www.youtube.com/watch?v=tFRXsngz4UQ.

** [Video] “Hypothesis tests, p-value”**
from Statistics Learning Center (Dr. Nic). 2011.
www.youtube.com/watch?v=0zZYBALbZgg.

*[Video]*** “Understanding
the p-value”** from Statistics Learning Center (Dr. Nic). 2011.

www.youtube.com/watch?v=eyknGvncKLw.

** [Video] “Important statistical concepts:
significance, strength, association, causation”** from

**Statistics Learning Center (Dr. Nic). 2012. www.youtube.com/watch?v=FG7xnWmZlPE.**

**
“Understanding statistical inference”** from Dr. Nic. 2015. Learn
and Teach Statistics & Operations Research.
learnandteachstatistics.wordpress.com/2015/11/09/understanding-statistical-inference/.

** “Basic concepts of hypothesis testing”**
in McDonald, J.H. 2014.

*Handbook of Biological Statistics*. www.biostathandbook.com/hypothesistesting.html.

** “Hypothesis testing”**, section 4.3,
in Diez, D.M., C.D. Barr , and M. Çetinkaya-Rundel. 2012.

*OpenIntro Statistics*, 2nd ed. www.openintro.org/.

*“Hypothesis Testing with One Sample”,*** sections 9.1–9.2 **in Openstax. 2013.

*Introductory Statistics*. openstaxcollege.org/textbooks/introductory-statistics.

** "Proving causation"** from Dr. Nic.
2013. Learn and Teach Statistics & Operations Research.
learnandteachstatistics.wordpress.com/2013/10/21/proving-causation/.

*[Video]*** “Variation and
Sampling Error”** from Statistics Learning Center (Dr. Nic). 2014.
www.youtube.com/watch?v=y3A0lUkpAko.

*[Video]*** “Sampling: Simple
Random, Convenience, systematic, cluster, stratified”** from

**Statistics Learning Center (Dr. Nic). 2012. www.youtube.com/watch?v=be9e-Q-jC-0.**

** “Confounding variables”** in McDonald,
J.H. 2014.

*Handbook of Biological Statistics*. www.biostathandbook.com/confounding.html.

** “Overview of data collection principles”**,
section 1.3, in Diez, D.M., C.D. Barr , and M. Çetinkaya-Rundel. 2012.

*OpenIntro Statistics*, 2nd ed. www.openintro.org/.

** “Observational studies and sampling strategies”**,
section 1.4, in Diez, D.M., C.D. Barr , and M. Çetinkaya-Rundel. 2012.

*OpenIntro Statistics*, 2nd ed. www.openintro.org/.

** “Experiments”**, section 1.5, in Diez,
D.M., C.D. Barr , and M. Çetinkaya-Rundel. 2012.

*OpenIntro Statistics*, 2nd ed. www.openintro.org/.

* *

### Exercises F

1. Which of the following pair is the null hypothesis?

A) The number of heads from the coin is not different from the
number of tails.

B) The number of heads from the coin is different from the number of tails.

2. Which of the following pair is the null hypothesis?

A) The height of boys is different than the height of girls.

B) The height of boys is not different than the height of
girls.

3. Which of the following pair is the null hypothesis?

A) There is an association between classroom and sex. That is,
there is a difference in counts of girls and boys between the classes.

B) There is no association between classroom and sex. That is, there is no difference in counts of girls and boys between the classes.

4. We flip a coin 10 times and it lands on heads 7 times. We want to know if the coin is fair.

What is the null hypothesis?

Looking at the code below, and assuming an *alpha* of
0.05,

What do you decide (use the *reject* or *fail to reject*
language)?

In practical terms, what do you conclude?

binom.test(7, 10, 0.5)

Exact binomial test

number of successes = 7, number of trials = 10, p-value = 0.3438

5. We measure the height of 9 boys and 9 girls in a class, in centimeters. We want to know if one group is taller than the other.

What is the null hypothesis?

Looking at the code below, and assuming an *alpha* of
0.05,

What do you decide (use the *reject* or *fail to
reject* language)?

In practical terms, what do you conclude? Address the practical importance of the results.

Girls = c(152, 150, 140, 160, 145, 155, 150, 152, 147)

Boys = c(144, 142, 132, 152, 137, 147, 142, 144, 139)

t.test(Girls, Boys)

Welch Two Sample t-test

t = 2.9382, df = 16, p-value = 0.009645

mean of x mean of y

150.1111 142.1111

mean(Boys)

sd(Boys)

quantile(Boys)

mean(Girls)

sd(Girls)

quantile(Girls)

boxplot(cbind(Girls, Boys))

6. We count the number of boys and girls in two classrooms. We are interested to know if there is an association between the classrooms and the number of girls and boys. That is, does the proportion of boys and girls differ statistically across the two classrooms?

What is the null hypothesis?

Looking at the code below, and assuming an *alpha* of
0.05,

What do you decide (use the *reject* or *fail to
reject* language)?

In practical terms, what do you conclude?

Classroom__Girls__ __Boys__

A 13 7

B 5 15

Input =("

Classroom Girls Boys

A 13 7

B 5 15

")

Matrix = as.matrix(read.table(textConnection(Input),

header=TRUE,

row.names=1))

fisher.test(Matrix)

Fisher's Exact Test for Count Data

p-value = 0.02484

Matrix

rowSums(Matrix)

colSums(Matrix)

prop.table(Matrix,

margin=1)

### Proportions for each row

barplot(t(Matrix),

beside = TRUE,

legend = TRUE,

ylim = c(0, 25),

xlab = "Class",

ylab = "Count")