Biostatistics in Dentistry
Instructor: Chuck Spiekerman
Statistics – used to describe characteristics of a large
population using small subset (sample).
Example: U.S. presidential election poll
CNN/ORC Poll. May 29-31, 2012. N=895 registered voters nationwide.
Margin of error ± 3.5.
"Suppose that the presidential election were being held today and you
had to choose between Barack Obama as the Democratic Party's
candidate, and Mitt Romney as the Republican Party's candidate. Who
would you be more likely to vote for: Barack Obama, the Democrat, or
Mitt Romney, the Republican?"
5/29-31/12
Barack
Obama
%
49
Mitt
Romney
%
46
Neither
(vol.)
%
4
Other
(vol.)
%
1
From www.pollingreport.com
With careful random sampling, there’s a good chance
that the %’s in the sample will be close to the %’s in
the population of interest.
But the “answers” we get are random (because of the
random sampling). Each different sample is going
to give a different answer.
In Statistics we use what we know about the random
sampling, and combine that with probability theory to
make more precise statements about how accurate
our sample statistics are in terms of describing the
actual population parameters of interest.
Biostatistics in Dentistry
Biostatistics ≈ statistical methods for
applications in medical/dental research
Medical/Dental Research often involves
human subjects,
study of complex mechanisms.
This leads to highly variable outcomes.
Statistical methods are used in an attempt to
separate the effects of interest from random
“noise” in the data.
Once understood, statistical methods give a
good basis for understanding what can and
what can’t be inferred from the data that are
produced in medical/dental studies.
Example: Chewing Gum Study
 Fourth grade students (ten year olds)
recruited.
 Students assessed for DMFS at study entry.
 Students randomly assigned to various
chewing-gum regimens: no gum, sucrose
gum, xylitol gum, sorbitol gum or
xylitol/sorbitol gum.
 Students assessed again for DMFS 2 years
after study entry.
We will focus on two groups, those randomized
to gum “B” and to gum “C”.
We wish to evaluate whether the two groups of
children differ with respect to caries progression.
Chewing Gum Data
Group B
id #
420
830
835
847
853
855
856
858
859
867
868
873
876
879
881
886
888
890
896
897
903
908
913
915
919
920
946
950
951
952
958
959
966
967
1,002
before
3
12
0
2
7
0
1
0
1
3
0
7
3
3
7
5
0
7
11
3
0
5
8
6
6
3
2
2
3
7
4
0
2
1
8
DMFS
after
3
0
0
1
8
0
6
0
0
4
6
2
7
0
6
0
3
1
5
3
0
7
6
10
0
4
0
0
1
10
2
0
0
0
8
Group C
change
0
-12
0
-1
1
0
5
0
-1
1
6
-5
4
-3
-1
-5
3
-6
-6
0
0
2
-2
4
-6
1
-2
-2
-2
3
-2
0
-2
-1
0
id #
331
335
336
340
342
344
345
346
348
352
353
354
355
356
364
366
374
380
385
389
392
396
401
402
403
404
405
408
415
416
417
422
423
428
430
434
455
456
457
460
before
2
8
0
0
0
1
0
3
5
1
1
8
5
1
2
5
6
5
0
3
0
0
2
8
7
0
3
3
5
12
4
6
16
2
7
4
7
0
4
1
DMFS
after
8
9
1
0
0
1
5
2
5
2
6
10
6
8
2
8
11
19
2
2
0
3
1
15
18
1
8
4
8
15
12
0
18
7
5
10
8
0
12
0
change
6
1
1
0
0
0
5
-1
0
1
5
2
1
7
0
3
5
14
2
-1
0
3
-1
7
11
1
5
1
3
3
8
-6
2
5
-2
6
1
0
8
-1
We can get a better understanding of the data using
histograms.
Gum B
Gum C
-15
-10
-5
0
5
10
15
5
10
15
5
10
15
change in DMFS
Gum B
-15
-10
-5
0
Change in DMFS
Gum C
-15
-10
-5
0
Change in DMFS
Is it obvious that the two gums have different effects?
Another common method used to compare groups.
Boxplot
20
18
change in DMFS
10
51
0
-10
42
-20
N=
35
40
B
C
gum type
 The lines inside the boxes are medians (50th
percentiles)
 The top of boxes are the 75th percentiles
 The bottom of the boxes are the 25th percentiles
 The “whiskers” extend to the furthest
observations within 1.5 interquartile ranges
(distance between 25th and 75th percentiles) from
the boxes
 The dots represent further outlying observations
 To objectively compare two sets of continuous
data, we often focus on the sample means
within the groups.
change in DMFS
N
gum typ e
Mean
Std Dev iation
B
35
-.8 3
3.5 7
C
40
2.6 3
3.8 0
 The sample mean is the average of the values.
It is a measure of the “center” of the data.
sample mean (group B) =
0  12  ...  1  0
 .83
35
 The standard deviation is a measure of
variability of the data; or, how spread out they
are.
 The standard deviation can be loosely
interpreted as the average distance of the
observations from the sample mean.
change in DMFS
N
gum typ e
Mean
Std Dev iation
B
35
-.8 3
3.5 7
C
40
2.6 3
3.8 0
mean
Gum B
< 1 SD
1 SD >
><
mean
Gum C
<
-15
-10
-5
1 SD
><
0
1 SD
5
>
10
15
change in DMFS
Non-standard graphical representation of means
and standard deviations
Summation Notation
We will use summation notation often.
n
x
i 1
i
 x1  x 2    x n
Example:
Suppose x1 = 5, x2 = -4, and x3 = 7, then
3
x
i 1
i
 x1  x2  x3  5  4  7  8
.
In summation notation, the sample mean, X , is
written
X
n
1
n
x
i 1
i
.
The sample variance, s2, is written
s 
2
 x
n
1
n 1
i 1
i
 X
2
,
2
s

s
and the sample standard deviation is
.
Back to the Data: Which are we seeing?
A. The two groups are systematically different.
The difference is reflected in the observed
mean changes in DMFS.
-- or -B. The two groups are basically the same. The
difference we observed in sample means is due
to the random sampling.
 In Hypothesis Testing, we go about answering
this question by assuming that the second of these
alternatives is the truth.
 In this example we assume that option B is
correct. We assume that the two gums are
basically the same.
 Then we use probability theory to calculate how
likely it would be for the sample means to be as
different as observed in a trial where everybody
was chewing the same gum.
“Behavior” of the sample statistics
 We can predict what values are likely and
unlikely to be seen when sampling one
observation from the population using a very
large sample to compute a histogram.
 With respect to sample statistics, like the
sample mean, we usually only see one
observation.
 Luckily, probability theory gives us the
Central Limit Theorem, which basically says
that averages (like the sample mean) all
behave in a specific way.
 Knowing how the sample means should
behave will let us make informed judgments
on whether the observed statistics are more
consistent with coming from
o two different groups, or
o the same group with two different labels
Deciding whether there are really two
different gum groups
 First, we assume/pretend that the two gums have
equivalent effects on DMFS.
 Then, we use probability theory to estimate how
likely different outcomes with respect to different
mean DMFS would be.
Probability distribution of possible differences
in mean DMFS if gums are equivalent
-4
-3
-2
-1
0
1
2
3
4
Mean difference of DMFS change
Probability is represented by area under the curve.
 Finally, we look at the actual observed data and
see whether they are consistent with our assumed
distribution.
Output from the statistical computer package SPSS
Independent Samples Tes t
t-test fo r Equality of Means
t
change in
DMFS
df
Sig.
(2-tailed)
Mean
Differen ce
Std. Erro r
Differen ce
Equal variances
assumed
-4.04
73
.00 013
-3.45
.86
Equal variances
not assumed
-4.06
72.6
.00 013
-3.45
.85
The circled value is the p-value of the t-test for
equality of means.
The p-value is the estimate of the probability
that a difference of –3.45 DMFS would be seen
between groups if, in fact, the two gums were
equivalent.
Since the probability (.00013, about 1 in 8000)
is very low, this leads us to doubt our initial
assumption that the two groups could actually be
equivalent.
This is the basic procedure for a hypothesis test.
Estimating the difference
 Using the sample means we estimated that
one gum was associated with 3.45 less
DMFS, on average.
 This estimate would differ depending on
which 75 children we happened to pick.
 Our hypothesis test convinced us that the
two gums were not the same; that is, the
difference was not 0.
 We can use similar methodology find all
other possible differences that our data
would not contradict.
 A 95% confidence interval for the mean
difference in DMFS between group B and
group C is
(1.75 to 5.16).
 This contains all possible true differences
between groups that would have at least a
95% chance of producing our observed data.
This class will concentrate mostly on
- developing a “toolbox” of necessary
probability and statistical theory (Days
1-4)
- introducing methods of inference for
various types of data
o develop a useful statistic
o figure out distribution of statistic
(almost always related to Normal
distribution)
o use distribution to estimate
precision of statistic and perform
hypothesis tests