Statistical tests /certified fixed orthodontic courses by Indian dental academy

Statistical tests and their
relevance to Dental Research
INDIAN DENTAL ACADEMY
Leader in continuing dental education
www.indiandentalacademy.com

www.indiandentalacademy.com

Contents
 Introduction
 Descriptive statistics
 Tests of significance
 Parametric test
 Non Parametric test
 Application of tests in dental research
 Conclusion
 Bibliography

www.indiande
ntalacademy.c
om

Introduction
Normal BP – 120/80 mm Hg
Europeans are taller than Asians
Average male adult weighs 70kgs
Drug A is better than drug B

- Endless
Cannot be arrived by just Raw data
Numbers tell tales – Speak the language of
STATISTICS – Adds meaning to data – helps to
interpret data
Thus lending “significance” to the study

www.indiande
ntalacademy.c
om

Descriptive statistics
STATISTICS
– Is the science of compiling, classifying &
tabulating numerical data and expressing the
results in a mathematical or graphical form.
OR
Statistics is the study of methods & procedures
for collecting, classifying, summarizing &
analyzing data & for making scientific inferences
from such data.
www.indiande
- Prof P.V.Sukhatme
ntalacademy.c
om

BIOSTATISTICS
–Is the branch of statistics applied to
biological or medical sciences (biometry).
OR
- Is that branch of statistics concerned
with mathematical facts and data relating
to biological events.
www.indiande
ntalacademy.c
om

VARIABLE
– A general term for any feature of the
unit which is observed or measured.
FREQUENCY DISTRIBUTION
– Distribution showing the number of
observations or frequencies at different
values or within certain range of values of
the variable.
www.indiande
ntalacademy.c
om

Mean
Dividing the total of all observations by
the number of observations
x1 + x 2 + x 3 + .... + xn ∑ x
x=
=
n
n
Eg. calculate the mean of DMFT scores 2.3, 2.0,2.7,3.0,2.0.
2.3 + 2.0 + 2.7 + 3.0 + 2.0 12
x=
=
= 2 .4
5
5
www.indiande
ntalacademy.c
om

Geometric mean (GM) – nth root of the
product
∑log x
n
GM =

( x1)( x 2 )( x3 )...( xn ) =

n
When the variation between the lowest and the highest
value is very high, geometric mean is advised & preferred

Harmonic mean (HM) – is the reciprocal of
the arithmetic mean of the reciprocal of the observations

1

n
HM =
=
1
1
1
∑
∑
n
xi
xi

www.indiande
ntalacademy.c
om

Median
– is the middle value, which divides the
observed values into two equal parts, when the
values are arranged in ascending or descending
order
 n + 1
 2 


Eg. calculate the median of DMFT scores 2.3, 2.0,2.7,3.0,2.0.
arrange in asc order,
5 +1 = 6
rd value
www.indiande
2.0,2.0.2.3,2.7,3.0 =
=3
= ie 2.3
ntalacademy.c
2
om

Mode
– is the value of the variable which occurs
most frequently
Mode = (3median – 2mean)
Eg. calculate the mode of DMFT scores 2.3, 2.0,2.7,3.0,2.0.
Mode = 2.0
Mode = 3 × 2.3 − 2 × 2.4 = 2.1
www.indiande
ntalacademy.c
om

Variance
Is the appropriate measure of dispersion
for interval or ratio level data
Computes how far each score is from the
mean
This is done by ( x − x )
Each score will have a deviation from the mean, so to
find the average deviation => we have to add all the
deviations and divide it by number of scores (just like
calculating mean)
www.indiande
ntalacademy.c
om

∑( x − x)
i.e.
N

but....∑ ( x − x ) = 0
So to eliminate this zero, square the deviations which
eliminates the (-) sign

i.e.

( x − x)2
∑
N

= S2

- is the average of the squared deviations
www.indiande
ntalacademy.c
om

Standard deviation(Root Mean
Square deviation)

 Is defined as the square root of the
arithmetic mean of the squared
deviations of the individual values from
their arithmetic mean
SD = (σ ) =

∑( x − x )
N −1

SD = ( s ) =

∑( x − x )
N

2

2

For small samples

For large samples
www.indiande
ntalacademy.c
om

For frequency distribution

SD = (σ ) =
SD = ( s ) =

∑ f ( x −x)
N −1
∑ f ( x −x)
N

2

2

For small samples

For large samples

www.indiande
ntalacademy.c
om

 Uses of SD
1. Summarizes the deviations of a large distribution from mean in
one figure used as unit of freedom
2. Indicates whether the variation from the mean is by chance or
real
3. Helps finding standard error- which determines whether the
difference b/n means of two samples is by chance or real
4. Helps finding the suitable size of the sample for valid
conclusions

www.indiande
ntalacademy.c
om

Standard error
Standard deviation of mean values
Used to compare means with one
another

Standard deviation SD
SE =
=
sample size
n

www.indiande
ntalacademy.c
om

Coefficient of variation
is a measure used to measure relative
variability
I.e,
Variation of same character in two or more different
series .
(eg – pulse rate in young & old person)
Variation of two different characters in one & same
series .
(eg – height & weight in same individual)

Standard Deviation
CV =
×100
Mean

www.indiande
ntalacademy.c
om

Normal curve and distribution
The histogram of the same frequency
distribution of heights, with large
number of observations & small class
intervals – gives a frequency curve which
is symmetrical in nature  Normal curve
or Gaussian curve

www.indiande
ntalacademy.c
om

Normal curve

x
www.indiande
ntalacademy.c
om

Characteristics of normal curve
Bell shaped
Symmetrical
Mean, Mode & Median – coincide
Has two inflections – the central part is convex, while
at the point of inflection the curve changes from
convexity to concavity
www.indiande
ntalacademy.c
om

 On preparing frequency distribution
with small class intervals of the data
collected, we can observe
1. Some observations are above the mean & others
are below the mean
2. If arranged in order, maximum number of frequencies
are seen in the middle around the mean & fewer at the
extremes decreasing smoothly
3. Normally half the observations lie above & half below
the mean & all are symmetrically distributed on each
side of mean

A distribution of this nature or shape is
www.indiande
called Normal or Gaussian distribution
ntalacademy.c
om

Arithmetically
mean ± 1SD limits , include 68.27% observations
mean ± 2SD limits, include 95.45% observations
mean ± 1.96 SD limits, include 95% observations
mean ± 3SDlimits, includes 99.73% observations

mean ± 2.58SDlimits, includes 99% observations
www.indiande
ntalacademy.c
om

Normal curve and distribution

68.26%
95.46%

x − 1SD
x − 2 SD

x

x + 1SD
x + 2 SD

99.72%

x − 3SD

x + 3SD
www.indiande
ntalacademy.c
om

Height
in cm
142.5
145.0
147.5
150.0
152.5
155.0
157.5
160.0(M)
162.5
165.0
167.5
170.0
172.5
175.0-177.5

frequency of
each group
3
8
15
45
90
155
194
195
136
93
42
16
6
2

Mean – 160.0

Mean
±1SD
680
68%

frequency with in
height limits of

Mean
±2SD
950
95%

SD – 5cm

Mean
±3SD
995
99%

www.indiande
ntalacademy.c
om

Relative or standard normal deviate
or variate – (Z)
Is the deviation from the mean in a
normal distribution or curve
observation - mean x − x
Z=
=
SD
SD
-in standard normal curve
the mean is taken as zero & SD as unity of one
www.indiande
ntalacademy.c
om

Skewness
 Skewness – as the static to measure the
asymmetry
 coefficient of skewness is 0
Positively (right) skewed
Negatively (left) skewed

Bimodal

www.indiande
ntalacademy.c
om

kurtosis
Kurtosis – is a measure of height of the
distribution curve
Coefficient of kurtosis is 3
Leptokurtic(high)

Platykurtic (flat)
Mesokurtic (normal)

www.indiande
ntalacademy.c
om

Tests of significance
Population
is any finite collection of elements
I.e – individuals, items, observations etc,.

Sample –
is a part or subset of the population

Parameter –
 is a constant describing a population

Statistic –
is a quantity describing a sample, namely a function
www.indiande
ntalacademy.c
of observations
om

Statistic
(Greek)

Parameter
(Latin)

Mean

x

µ

Standard
Deviation

s

σ

Variance

s2

σ

Correlation
coefficient

r

ρ

Number of
subjects

n

N

2

www.indiande
ntalacademy.c
om

Hypothesis testing
 Hypothesis H
 is an assumption about the status of a phenomenon or is a
statement about the parameters or form of population

Null hypothesis or hypothesis of no difference –
States no difference between statistic of a sample & parameter
of population or b/n statistics of two samples
This nullifies the claim that the experiment result is different
from or better than the one observed already
Denoted by

H0

www.indiande
ntalacademy.c
om

Alternate hypothesis –
Any hypothesis alternate to null hypothesis, which is to
be tested
Denoted by H1

Note : the alternate hypothesis is accepted when
null hypothesis is rejected
www.indiande
ntalacademy.c
om

Type I & type II errors
H 0 Accept
H0

is true

H1 is true

H1 Accept

No error

Type I error

Type II error

No error

Type I error = α
Type II error = β
When primary concern of the test is to see
whether the null hypothesis can be rejected
such test is called Test of significance
www.indiande
ntalacademy.c
om

 The probability of committing type I error is
called “P” value
 Thus p-value is the chance that the presence
of difference is concluded when actually there
is none
Type I error – important- fixed in advance at a
low level – such upper limit of tolerance of the
chance of type I error is called
Level of Significance (α)
Thus α is the maximum tolerable probability
www.indiande
of type I error
ntalacademy.c
om

Difference b/n level of significance & Pvalue LOS

1) Maximum tolerable
chance of type I error
2) α is fixed in advance

P-value

1) Actual probability of
type I error
2) calculated on basis of data
following procedures

The P-value can be more than α or less than α depending on data
When P-value is < than α  results is statistically significant
www.indiande
ntalacademy.c
om

The level of significance is usually
fixed at 5% (0.05) or 1% (0.01) or
0.1% (0.001) or 0.5% (0.005)
Maximum desirable is 5% level
When P-value is b/n
0.05-0.01 = statistically significant
< than 0.01= highly statistically significant
Lower than 0.001 or 0.005 = very highly significant
www.indiande
ntalacademy.c
om

Sampling Distribution
Zone of
Rejection H0
1/2 α

Zone of
Rejection H0
1/2 α

Zone of
Acceptance H0

x − 1.96 SD

x

x + 1.96SD

Confidence interval

Confidence limits – 95%
www.indiande
ntalacademy.c
om

Tests of significance
 Are mathematical methods by which
the probability (P) or relative frequency
of an observed difference, occurring by
chance is found
 Steps & procedure of test of
significance –
1. State null hypothesis H 0
2. State alternate hypothesis H1
3. Selection of the appropriate test to be utilized &
calculation of test criterion based on type of test
www.indiande
ntalacademy.c
om

4. Fixation of level of significance
5. Select the table & compare the calculated value with
the critical value of the table
6. If calculated value is > table value, H 0 is rejected
7. If calculated value is < table value, H 0is accepted
8. Draw conclusions
www.indiande
ntalacademy.c
om

TESTS IN TEST OF SIGNIFICANCE
Parametric
(normal distribution &
Normal curve )
Quantitative data
Student ‘t’ test
1) Paired
2) Unpaired
Z test
(for large samples)
One way ANOVA
Two way ANOVA

Non-parametric
(not follow
normal distribution)

Qualitative data
1) Z – prop test Qualitative
(quantitative converted
to qualitative )
1. Mann Whitney U test
2. Wilcoxon rank test
3. Kruskal wallis test
4. Spearman test
www.indiande
ntalacademy.c
om

Parametric

Uses

Non-parametric

Paired t test

Test of diff b/n
Paired observation

Wilcoxon signed
rank test

Two sample t test

Comparison of two
groups

Wilcoxon rank sum test
Mann Whitney U test
Kendall’s s test

One way Anova

Comparison of
several groups

Kruskal wallis test

Two way Anova

Comparison of groups Friedmann test
values on two variables
Spearman’s rank
Correlation
Measure of association Correlation
coefficient
B/n two variable
Kendall’s rank
correlation www.indiande
ntalacademy.c
Normal test (Z test )
Chi square test
om

Student ‘t’ test
Small samples do not follow normal
distribution as the large ones do => will
not give correct results
Prof W.S.Gossett – Student‘t’ test – pen
name – student
It is the ratio of observed difference b/n
two mean of small samples to the SE of
difference in the same
www.indiande
ntalacademy.c
om

Unpaired ‘t’ test
Types

Paired ‘t’ test

Actually, t-value  Z-value of large
samples, but the probability (P) of this is
determined by reference ‘t’ table
Degree of freedom (df)- is the quantity in
the denominator which is one less than
independent number of observations in a
sample
For unpaired ‘t’ test = n1 + n2 − 2
For paired ‘t’ test = n-1
www.indiande
ntalacademy.c
om

 Criteria for applying ‘t’ test –





Random samples
Quantitative data
Variable follow normal distribution
Sample size less than 30

 Application of ‘t’ test –
1. Two means of small independent sample
2. Sample mean and population mean
3. Two proportions of small independent samples
www.indiande
ntalacademy.c
om

Unpaired ‘t’ test
I)

Difference b/n means of two independent
samples
Data –
Group 1 Group 2
Sample size
Mean
SD

n1
x1

n2
x2

SD1

SD2

1) Null hypothesis H 0 ⇒ ( x1 − x2 ) = 0
2) Alternate hypothesis H1 ⇒ ( x1 − x2 ) ≠ 0
www.indiande
ntalacademy.c
om

3) Test criterion

t=

x1 − x2

SE ( x1 − x2 )

here SE of ( x1 − x2 ) is calculated by

1 1
SE of ( x1 − x2 ) = SD  + 
n n 
2 
 1

where SD =

SE ( x1 − x2 ) =

( n1 −1) SD12 + ( n2 −1) SD22
n1 + n2 − 2

( n1 − 1) SD12 + ( n2 − 1) SD22  1

n1 + n2 − 2

1
n + n 

2 
 1

www.indiande
ntalacademy.c
om

4) Calculate degree of freedom

df = ( n1 − 1) + ( n2 − 1) = n1 + n2 − 1

5) Compare the calculated value &
the table value
6) Draw conclusions

www.indiande
ntalacademy.c
om

Example – difference b/n caries experience of
high & low socioeconomic group
Sl
no

Details

High socio
economic group

Low socio
economic group

I

Sample size

n1 = 15

n2 = 10

II

DMFT

x1 = 2.91

x2 = 2.26

III

Standard deviation

SD1 = 0.27

SD2 = 0.22

x1 − x2

SE ( x1 − x2 )

0.65
=
= 6.34,
0.1027

t0.001 = 3.76

∴tc > t0.001

t=

There is a significant difference

df = 23

www.indiande
ntalacademy.c
om

T table

www.indiande
ntalacademy.c
om

Other applications
II) Difference b/n sample mean & population
x−µ
mean
t=
df = n − 1
SE = SD
n
III) Difference b/n two sample proportions
t=

p1 − p2
1 1
PQ + 
n n 
2 
 1

n1 p1 + n2 p2
where P =
n1 + n2
Q = 1− P

df = n1 + n2 − 2 www.indiande
ntalacademy.c
om

Paired ‘t’ test
 Is applied to paired data of observations
from one sample only when each individual
gives a paired of observations
 Here the pair of observations are correlated
and not independent, so for application of ‘t’
test following procedure is used1.
2.
3.

Find the difference for each pair y1 − y2 = x
Calculate the mean of the difference (x) ie x
Calculate the SD of the differences & later SE

 SD 
SE =  
 n

www.indiande
ntalacademy.c
om

4. Test criterion

x −0
x
t=
=
SE ( d ) SD( x )

n

5. Degree of freedom df = n − 1
6. Refer ‘t’ table & find the probability
of calculated value
7. Draw conclusions
www.indiande
ntalacademy.c
om

Example – to find out if there is any significant
improvement in DAI scores before and after orthodontic
treatment
Sl no

DAI before

DAI after

Difference

Squares

1

30

24

6

36

2

26

23

3

9

3

27

24

3

9

4

35

25

10

100

5

25

23

2

4

20

158

Total

www.indiande
ntalacademy.c
om

∑ x = 20 = 4
Mean x =
n

5

sum of squares, ∑ ( x − x ) = ( 6 − 4 ) + ( 3 − 4 ) + ( 3 − 4 ) + (10 − 4 ) + ( 2 − 4 )
2

SD =

( x − x )2
∑

2

2

2

= 4 + 1 + 1 + 36 + 4
= 46

=

46
= 11.5 = 3.391
4

n −1
SD 3.391
∴SE =
=
=1.5179
n
5
x
4
∴c =
t
=
= 2.6352
SE 1.5179
but t 0.5 = 2.78
∴ c < t 0.5
t

2

Hence not significant

df = n −1 = 4
www.indiande
ntalacademy.c
om

2

Z test (Normal test)
Similar to ‘t’ test in all aspect except that
the sample size should be > 30
In case of normal distribution, the
tabulated value of Z at -

5% level = Z 0.05 = 1.960
1% level = Z 0.01 = 2.576
0.1% level = Z 0.001 = 3.290
www.indiande
ntalacademy.c
om

 Z test can be used for –
1. Comparison of means of two samples –

x1 − x2
Z=
SE ( x1 − x2 )

where SE ( x1 − x2 ) =

( SE

1

2

+ SE2

2

)

2
 SD12 SD2 
= 
 n + n 

2 
 1

2. Comparison of sample mean & population mean

Z =

x −µ
SD 2
n

www.indiande
ntalacademy.c
om

3. Difference b/n two sample proportions
Z=

p1 − p2

1
1 
PQ + 
n
n2 
1




n1 p1 + n2 p2
where P =
n1 + n2

Q = 1− P

4. Comparison of sample proportion
(or percentage) with population proportion
(or percentage)
Z=

p−P
  1 
 PQ 
  n 

Where p = sample proportion
P = populn proportion
www.indiande
ntalacademy.c
om

Analysis of variance (ANOVA)
 Useful for comparison of means of several
groups
 Is an extension of student’s ‘t’ test for more
than two groups
 R A Fisher in 1920’s
 Has four models
1.
2.
3.
4.

One way classification (one way ANOVA )
Single factor repeated measures design
Nested or hierarchical design
Two way classification (two way ANOVA)
www.indiande
ntalacademy.c
om

One way ANOVA
Can be used to compare likeEffect of different treatment modalities
Effect of different obturation techniques on the apical
seal , etc,.

www.indiande
ntalacademy.c
om

1

Groups (or treatments)

2

Individual values

i

k

x11
x12

x21

xi1

xk 1

x22

xi 2

x1n

x2 n

xin

xk 2
xkn

n

n

n

n

T2

Ti

Tk

S2
x2

Si

Sk

xi

www.indiande
k
ntalacademy.c
om

Calculate
No of observations
Sum of x values
Sum of squares
Mean of values

Τ1 = x11 + x12 + ... + x1n
S1 = ( x11 ) + ( x12 ) + .. + ( x1n )
2

x1 =

2

T1
n

2

x

ANOVA table
Sl Source
Degree Sum of squares
no of
of
variation freedom
I

Between
Groups

II

With in
groups

III Total

k − 1 ∑i ( xi − x )
n − k ∑ ∑ (x
i

ij

j

n −1 ∑ ∑ (x
i

j

ij

2

− xi )

Mean sum of
squares

F ratio or
variance ratio


2
T2 
2
=  ∑i xi −  2 ∑i ( xi − x )

N  SB =


k −1
2

2
SB
( k − 1, N − k )
2
SW

T 2 
xij2 − ∑ i  i 
T
∑i∑ j
2
= ∑i ∑ j xij −∑i i 2
 ni 
SW =
ni
N−k
2

T
2
2
− x ) = ∑i ∑ j xij −
N


2






ST2 =

∑∑
i

T2
x −

2
j ij

N −1

N

www.indiande
ntalacademy.c
om

ANOVA

www.indiande
ntalacademy.c
om

Example- see whether there is a difference in number of
patients seen in a given period by practitioners in three
group practice
Practice
Individual values

A

C

B

268

387

161

349

264

346

328

423

324

209

254

293

292

239

Calculate
No of observations (n)

5

4

5

Sum of x values

1441

1328

1363

Sum of squares

426899

462910

393583

Mean of values

288.2

332.0

www.indiande
272.6
ntalacademy.c
om

Between group sum of squares

( ∑ x ) + ( ∑ x ) + ( ∑ x ) − ( ∑ x + ∑ x +∑ x )
=
2

2

A

2

B

nA

C

nB

A

B

2

C

n A + nB + nC

nC

= 8215.71

Total sum of squares
= ∑x + x + x
2
A

2
B

2
C

(∑ x
−

A

+ xB + xC )

2

n A + nB + nC

= 63861.71

With in group sum of squares
= total SS - between SS
= 55646.0

www.indiande
ntalacademy.c
om

ANOVA table
Source of Degree of
variation freedom

I

Between
Groups

3 −1 = 2

II

With in
groups

14 − 3 = 11

III

Total

14 − 1 = 13

F = 0.81

Sum of squares Mean sum of squares F ratio or variance
ratio

8215.71

8215.71
4107.86
= 4107.86
= 0.81
2
5088.73
55646
= 5088.73
11

55646.0

Sl
no

63861.71
F0.05 = 3.98 df = ( 2,11)

Because FC < FT , there is no significant difference
in the number of patients attending 3 different practice

www.indiande
ntalacademy.c
om

Further, any particular pair of
treatments can be compared using
SE of difference b/n two means
Eg – xd & xc
 1 1
SE ( xd − xc ) = MSE  + 
n
nc 
 d

& difference xd − xc may be tested by using
' t' test criterion

xd − xc
t=
SE ( xd − xc )

www.indiande
ntalacademy.c
om

Two way ANOVA
Is used to study the impact of two
factors on variations in a specific
variable
Eg – Effect of age and sex on DMFT value

www.indiande
ntalacademy.c
om

Sample values
blocks

Treatments

i

x11 x21

x31

ii
..

x12 x22

n
Sample
size
Total
Mean
value

sample size

Total

xk 1

k

T1

x1

x32

xk 2

k

T2

x2

x1n x2 n

x3n

xkn

k

Tn

xn

n n

n

n

nk = N

T1 T2

T3

Tk

x1

x3

xk

x2

Mean
value

T

x

www.indiande
ntalacademy.c
om

2 way ANOVA table
Sl
no

Source

I

Blocks

II

Treatments

III Residual
or error
IV

Sum of
squares

SSblocks
SStreatments

Degree of
freedom

n −1
k −1

SS residual ( n − 1)( k − 1)

Total

SStotal

Mean sum of squares
(MSS)

MS blocks

SS blocks
=
n −1

Variance ratio
F

F1 =

MSblocks
MS residual

SStreatments F = MS treatment
MStreatments =
2
MS residual
k −1
SS residual
MS residual =
( n − 1)( k − 1)

( nk − 1) = N − 1

F1 = variance ratio of blocks with df of ( n − 1) Vs ( n − 1)( k − 1)
www.indiande
ntalacademy.c
F2 = variance ratio of treatment' s with df of ( k − 1) Vs ( n − 1)( k − 1)
om

Multiple comparison tests
1. Fisher’s procedure – student’s ‘t’ test
2. Least significant difference method (LSD)
 Just like student’s ‘t’ test
 To test significant difference b/n two groups or variable
means

1. Scheffe’s significant difference procedure
 Is applicable when groups having heterogeneous variance
or variations

1. Tukey’s method
 For comparison of the differences b/n all possible pairs of
treatments or group means
www.indiande
ntalacademy.c
om

5. Duncan’s multiple comparison test
–

For all comparisons of paired groups only

6. Dunnet’s comparison test procedure
–

For comparison of one control and several treatment
groups

www.indiande
ntalacademy.c
om

Non parametric tests
 Here the distribution do not require any
specific pattern of distribution. They are
applicable to almost all kinds of distribution
 Chi square test
 Mann Whitney U test
 Wilcoxon signed rank test
 Wilcoxon rank sum test
 Kendall’s S test
 Kruskal wallis test
 Spearman’s rank correlation
www.indiande
ntalacademy.c
om

Chi square test
 By Karl Pearson & denoted as χ²
 Application
1. Alternate test to find the significance of difference in
two or more than two proportions
2. As a test of association b/n two events in binomial or
multinomial samples
3. As a test of goodness of fit

www.indiande
ntalacademy.c
om

Requirement to apply chi square test
Random samples
Qualitative data
Lowest observed frequency not less than 5

Contingency table
Frequency table where sample classified according
to two different attributes
2 rows ; 2 columns => 2 X 2 contingency table
r rows : c columns => rXc contingency table

χ =∑
2

(O − E)
E

2

O – observed frequency
E – expected frequency
www.indiande
ntalacademy.c
om

 Steps
1. State null & alternate hypothesis
2. Make contingency table of the data

( r × c)

3. Determine expected frequency by
r×c
E=
N ( total frequency)
4. Calculate chi-square of each by-

χ

2

(O − E)
=
E

2
www.indiande
ntalacademy.c
om

5. calculate degree of freedom

df = ( c −1)( r −1)

6. Sum all the chi-square of each cell – this gives
chi-square value of the data

χ =∑
2

(O − E)

2

E

7. Compare the calculated value with the table
value at any LOS
8. Draw conclusions

www.indiande
ntalacademy.c
om

 Example from a dental health campaign
School

Oral hygiene

Total

G

F-

P

Below avg

62
(85.9)

103
(93.0)

57
(45.2)

11
(8.9)

233

Avg

50
(43.9)

36
(47.5)

26
(23.1)

7
(4.6)

119

Above avg

80
(62.3)

69
(67.5)

18
(32.8)

2
(6.5)

169

Total

E=

F+

192

208

101

20

521

r×c
N ( total frequency)

χ c2 = ∑

(O − E)2
E

= 31.4

df = ( c −1)( r −1) = 2 × 3 = 6
Table χ t2 at P = 0.001 is 22.46

Hence significant difference

www.indiande
ntalacademy.c
om

Alternate formulae
If we have contingency table
a
c
a+c

b
d
b+d

a+b
c+d
a+b+c+d=N

N ( ad − bc )
χ =
with df = 1
( a + b )( c + d )( a + c )( b + d )
2

2

If one of the value is below 5 => Yate’s
correction formula

2


N
N  ad − bc − 
2

χ2 =
( a + b )( c + d )( a + c )( b + d )

with df = 1

www.indiande
ntalacademy.c
om

 If the table is larger than 2X2, Yate’s correction cannot
be applied – then the small frequency (<5) can be
pooled or combined with next group or class in the
table
 Chi square test only tells the presence or absence of
association, but does not measure the strength of
association
www.indiande
ntalacademy.c
om

 If degree of association as to be
calculated then –
ad − bc
1. Yule’s coefficient of association Q =
ad + bc
Y=

2. Yule’s coefficient of colligation
3. -

ad − bc
V=
( a + b )( a + c )( b + d )( c + d )

1 − bc

ad

1 + bc

ad

4. Pearson’s coefficient of contingency
C=

χ2
N + χ2

www.indiande
ntalacademy.c
om

Wilcoxon signed rank test
Is equivalent to paired ‘t’ test
Steps
Exclude any differences which are zero
Put the remaining differences in ascending order,
ignoring the signs
Gives ranks from lowest to highest
If any differences are equal, then average their ranks
Count all the ranks of positive differences – T+
Count all the ranks of negative differences – Twww.indiande
ntalacademy.c
om

If there is no differences b/n variables then T+
& T_ will be similar, but if there is difference
then one sum will be large and the other will be
much smaller
T= smaller of T+&T_
Compare the T value with the critical value for
5%, 2% & 1% significance level
A result is significant if it is smaller than
critical value
www.indiande
ntalacademy.c
om

Example:Results of a placebo-controlled clinical trail to test
the effectiveness of sleeping drug
Patients
1
2
3
4
5
6
7
8
9
10

Sleep hrs
Drug
Placebo
6.1
5.2
7.0
7.9
8.2
3.9
7.6
4.7
6.5
5.3
8.4
5.4
6.9
4.2
6.7
6.1
7.4
3.8
5.8
6.3

Difference
0.9
-0.9
4.3
2.9
1.2
3.0
2.7
0.6
3.6
-0.5

Rank with signs
+
3.5
-3.5
10
7
5
8
6
2
9
-1
50.5
-4.5
www.indiande
ntalacademy.c
om

Calculated T=-4.5 df =10,
Table value at 5% (n=10)= 8
Cal T< table value, H0 is rejected
We conclude that sleeping drug is more
effective than the placebo
www.indiande
ntalacademy.c
om

Mann Whitney U test
Is used to determine whether two
independent sample have been drawn
from same sample
It is a alternative to student ‘t’ test &
requires at least ordinal or normal
measurement
n1 ( n1 + 1)
U = n1n2 +
− R1 or R2
2
Where, n1n2 are sample sizes
R1 R2 are sum of ranks assigned to I & II group

www.indiande
ntalacademy.c
om

Procedure
All the observation in two samples are ranked
numerically from smallest to largest without
regarding the groups
Then identify the observation for I and II samples
Sum of ranks for I and II sample determined
separately
Take difference of two sum T =R1 - R2

www.indiande
ntalacademy.c
om

Comparison of birth weights of children born to 15 non
smokers with those of children born to 14 heavy smokers
NS 3.9 3.7 3.6 3.7 3.2 4.2 4.0 3.6 3.8 3.3 4.1 3.2 3.5 3.5 2.7
HS 3.1 2.8 2.9 3.2 3.8 3.5 3.2 2.7 3.6 3.7 3.6 2.3 2.3 3.6

Ranks assignments
R1

26

23

16

21

8

29

27

17

24

12

28

10

15

13

R2

7

5

6

11

25

14

9

4

20

22

19

2

1

03

18
www.indiande
ntalacademy.c
om

Sum of R1= 272 and Sum of R2=163
Difference T=R1 – R2 is 109

The table value of T0.05 is 96 , so reject the H0

We conclude that weights of children born to the
heavy smokers are significantly lower than those of
the children born to the non-smokers (p<0.05)

www.indiande
ntalacademy.c
om

Applications of statistical tests
in Research Methods

www.indiande
ntalacademy.c
om

One variable,
two group sample problem

www.indiande
ntalacademy.c
om

One variable,
multiple group sample problem

www.indiande
ntalacademy.c
om

Two variable problem
Interested in relationship
b/n two variables
Use linear
regression

Yes

Are both continuous variables
No

Is one variable continuous
& other categorical
No

Yes

Use analysis
of variance
Or
Logistic regression

Both variables
categorical
Use contingency
table analysis

www.indiande
ntalacademy.c
om

Multiple – variable problem

Research interested in relationship
B/n more than two variables

Use multiple regression
Or
Multivariate analysis
www.indiande
ntalacademy.c
om

Conclusion
Statistics are excellent tools in research data
analysis; how ever, if inappropriately used they may
make the results of a well conducted research study
un-interpretable or meaningless

www.indiande
ntalacademy.c
om

Bibliography
 Biostatistics
– Rao K Vishweswara, Ist edition.
 Methods in Biostatistics
– Dr Mahajan B K, 5th edition.
 Essentials of Medical Statistics
– Kirkwood Betty R, 1st edition.
 Health Research design and Methodology
– Okolo Eucharia Nnadi.
 Simple Biostaistics
– Indrayan,1st edition.
 Statistics in Dentistry
www.indiande
ntalacademy.c
om
– Bulman J S

Statistical tests /certified fixed orthodontic courses by Indian dental academy

More Related Content

What's hot (20)

Similar to Statistical tests /certified fixed orthodontic courses by Indian dental academy (20)

More from Indian dental academy (20)

Statistical tests /certified fixed orthodontic courses by Indian dental academy