Introduction to Statistics with GraphPad

Prism 8
Anne Segonds-Pichon
v2019-03
Outline of the course

• Power analysis with G*Power

• Basic structure of a GraphPad Prism project

• Analysis of qualitative data:

• Chi-square test

• Analysis of quantitative data:

• Student’s t-test, One-way ANOVA, correlation and curve fitting
 Power analysis
• Definition of power: probability that a statistical test will reject a false null hypothesis (H0).
• Translation: the probability of detecting an effect, given that the effect is really there.

• In a nutshell: the bigger the experiment (big sample size), the bigger the power (more likely
to pick up a difference).
• Main output of a power analysis:
• Estimation of an appropriate sample size
• Too big: waste of resources,
• Too small: may miss the effect (p>0.05)+ waste of resources,
• Grants: justification of sample size,
• Publications: reviewers ask for power calculation evidence,
• Home office: the 3 Rs: Replacement, Reduction and Refinement.
Experimental design
Think stats!!
• Translate the hypothesis into statistical questions:
• What type of data?
• What statistical test ?
• What sample size?
• Very important: Difference between technical and biological replicates.
Technical Biological

n=1 n=3
 Power analysis

A power analysis depends on the relationship between 6 variables:

• the difference of biological interest

Effect size
• the variability in the data (standard deviation)
• the significance level (5%)
• the desired power of the experiment (80%)
• the sample size
• the alternative hypothesis (ie one or two-sided test)
1 The difference of biological interest
• This is to be determined scientifically, not statistically.
• minimum meaningful effect of biological relevance

• the larger the effect size, the smaller the experiment will need to be to detect it.
• How to determine it?
• Substantive knowledge, previous research, pilot study …

2 The Standard Deviation (SD)

• Variability of the data
• How to determine it?
• Substantive knowledge, previous research, pilot study …
• In ‘power context’: effect size: combination of both:
• e.g.: Cohen’s d = (Mean 1 – Mean 2)/Pooled SD
3 The significance level
• usually 5% (p<0.05), probability of the Type I error α
• p-value is the probability that a difference as big as the one observed could be found even
if there is no effect.
• Probability that an effect occurs by chance alone

• Don’t throw away a p-value=0.051 !

The significance level, critical value, α and β

• α : the threshold value that we measure p-values against.

• For results with 95% level of confidence: α = 0.05
• = probability of type I error
• p-value: probability that the observed statistic occurred by chance alone
• Statistical significance: comparison between α and the p-value
• p-value < 0.05: reject H0 and p-value > 0.05: fail to reject H0
The critical value
Example: 2-tailed t-test with n=15 (df=14)

T Distribution

0.95
0.025 0.025

t(14)
t=-2.1448 t=2.1448

• In hypothesis testing, a critical value is a point on the test distribution that is

compared to the test statistic to determine whether to reject the null hypothesis
• Example of test statistic: t-value
• If the absolute value of your test statistic is greater than the critical value, you can
declare statistical significance and reject the null hypothesis
• Example: t-value > critical t-value
4 The desired power: 80%

• Type II error (β) is the failure to reject a false H0

• Direct relationship between Power and type II error:

• if β = 0.2 and Power = 1 – β = 0.8 (80%)
• Hence a true difference will be missed 20% of the time
• General convention: 80% but could be more or less

• Cohen (1988):
• For most researchers: Type I errors are four times
more serious than Type II errors: 0.05 * 4 = 0.2
• Compromise: 2 groups comparisons: 90% = +30% sample size, 95% = +60%
 5 The sample size: the bigger the better?

• It takes huge samples to detect tiny differences but tiny samples to detect huge differences.

• What if the tiny difference is meaningless?

• Beware of overpower
• Nothing wrong with the stats: it is all about
interpretation of the results of the test.

• Remember the important first step of power analysis

• What is the effect size of biological interest?
6 The alternative hypothesis
• One-tailed or 2-tailed test? One-sided or 2-sided tests?

• Is the question:
• Is the there a difference?
• Is it bigger than or smaller than?

• Can rarely justify the use of a one-tailed test

• Two times easier to reach significance with a one-tailed than a two-tailed
• Suspicious reviewer!
To recapitulate:
• The null hypothesis (H0): H0 = no effect

• The aim of a statistical test is to reject or not H0.

Statistical decision True state of H0
H0 True (no effect) H0 False (effect)
Reject H0 Type I error α Correct
False Positive True Positive
Do not reject H0 Correct Type II error β
True Negative False Negative

• Traditionally, a test or a difference are said to be “significant” if the probability of type I

error is: α =< 0.05
 Hypothesis

Experimental design
Choice of a Statistical test

Power analysis: Sample size

Experiment(s)

Data exploration

Statistical analysis of the results

• Fix any five of the variables and a mathematical relationship can be used
to estimate the sixth.
e.g. What sample size do I need to have a 80% probability (power) to detect this particular
effect (difference and standard deviation) at a 5% significance level using a 2-sided test?
• Good news:
there are packages that can do the power analysis for you ... providing you have some prior
knowledge of the key parameters!
difference + standard deviation = effect size

• Free packages:
• R
• G*Power and InVivoStat
• Russ Lenth's power and sample-size page:
• http://www.divms.uiowa.edu/~rlenth/Power/

• Cheap package: StatMate (~ $95)

• Not so cheap package: MedCalc (~ $495)

Sample Statistical inference Population

Difference Meaningful? Yes

Real?

Statistical test

Big enough? Statistic

e.g. t, F …
=

Difference + Noise + Sample

Qualitative data
 Qualitative data
• = not numerical

• = values taken = usually names (also nominal)

• e.g. causes of death in hospital

• Values can be numbers but not numerical

• e.g. group number = numerical label but not unit of measurement

• Qualitative variable with intrinsic order in their categories = ordinal

• Particular case: qualitative variable with 2 categories: binary or dichotomous

• e.g. alive/dead or presence/absence
 Fisher’s exact and Chi2
Example: cats and dogs.xlsx
• Cats and dogs trained to line dance
• 2 different rewards: food or affection
• Question: Is there a difference between the rewards?

• Is there a significant relationship between the 2 variables?

– does the reward significantly affect the likelihood of dancing?

• To answer this type of question: Food Affection

– Contingency table Dance ? ?
No dance ? ?
– Fisher’s exact or Chi2 tests

But first: how many cats do we need?

G*Power
A priori Power Analysis
Example case:
Preliminary results from a pilot study on cats: 25% line-
danced after having received affection as a reward vs.
70% after having received food.

Power analysis with G*Power = 4 steps

Step1: choice of Test family

G*Power

Step 2 : choice of Statistical test

Fisher’s exact test or Chi-square for 2x2 tables

G*Power

Step 3: Type of power analysis

G*Power
Step 4: Choice of Parameters
Tricky bit: need information on the size of the
difference and the variability.
G*Power
Output:
If the values from the pilot study are good predictors and if you use a sample
of n=23 for each group, you will achieve a power of 83%.
 Chi-square and Fisher’s tests
• Chi2 test very easy to calculate by hand but Fisher’s very hard
• Many software will not perform a Fisher’s test on tables > 2x2
• Fisher’s test more accurate than Chi2 test on small samples
• Chi2 test more accurate than Fisher’s test on large samples
• Chi2 test assumptions:
• 2x2 table: no expected count <5
• Bigger tables: all expected > 1 and no more than 20% < 5
• Yates’s continuity correction
• All statistical tests work well when their assumptions are met
• When not: probability Type 1 error increases
• Solution: corrections that increase p-values
• Corrections are dangerous: no magic
• Probably best to avoid them
 Chi-square test

• In a chi-square test, the observed frequencies for two or more groups are compared with
expected frequencies by chance.

• With observed frequency = collected data

• Example with ‘cats and dogs’

 Chi-square test
Did they dance? * Type of Training * Animal Crosstabulation

Example: expected frequency of cats line dancing after having

Type of Training
Food as Affection as
Animal Reward Reward Total
Cat Did they
dance?
Yes Count
% within Did they dance?
26
81.3% 18.8%
6 32
100.0%
received food as a reward:
No Count 6 30 36

Total
% within Did they
Count
dance? 16.7%
32
83.3%
36
100.0%
68 Direct counts approach:
% within Did they dance? 47.1% 52.9% 100.0%
Dog Did they Yes Count 23 24 47
dance?
No
% within Did they
Count
dance? 48.9%
9
51.1%
10
100.0%
19
Expected frequency=(row total)*(column total)/grand total
Total
% within Did they
Count
dance? 47.4%
32
52.6%
34
100.0%
66
= 32*32/68 = 15.1
% within Did they dance? 48.5% 51.5% 100.0%

Probability approach:
Did they dance? * Type of Training * Animal Crosstabulation

Type of Training Probability of line dancing: 32/68

Probability of receiving food: 32/68
Food as Affection as
Animal Reward Reward Total
Cat Did they Yes Count 26 6 32
dance? Expected Count 15.1 16.9 32.0
No Count
Expected Count
6
16.9
30
19.1
36
36.0
Expected frequency:(32/68)*(32/68)=0.22: 22% of 68 = 15.1
Total Count 32 36 68
Expected Count 32.0 36.0 68.0
Dog Did they
dance?
Yes Count
Expected Count
23
22.8
24
24.2
47
47.0
For the cats:
No Count 9 10 19

Total
Expected Count
Count
9.2
32
9.8
34
19.0
66
Chi2 = (26-15.1)2/15.1 + (6-16.9)2/16.9 + (6-16.9)2 /16.9 + (30-19.1)2/19.1 = 28.4
Expected Count 32.0 34.0 66.0

Is 28.4 big enough for the test to be significant?

Results
 Fisher’s exact test: results
Dog Cat
30 30
Dance Yes Dance Yes
Dance No Dance No
20 20
Counts

Counts
10 10

0 0
Food Affection Food Affection

Cat Dog
100 100

• In our example: 80 80

cats are more likely to line dance if they are given food as

Percentage

Percentage
60 60 Dance No
Dance Yes
reward than affection (p<0.0001) whereas dogs don’t mind 40 40

(p>0.99).
20 20

0 0
Food Affection Food Affection
Quantitative data
 Quantitative data
• They take numerical values (units of measurement)

• Discrete: obtained by counting

• Example: number of students in a class
• values vary by finite specific steps
• or continuous: obtained by measuring
• Example: height of students in a class
• any values

• They can be described by a series of parameters:

• Mean, variance, standard deviation, standard error and confidence interval
 Measures of central tendency
Mode and Median

• Mode: most commonly occurring value in a distribution

• Median: value exactly in the middle of an ordered set of numbers

 Measures of central tendency
Mean

• Definition: average of all values in a column

• It can be considered as a model because it summaries the data

• Example: a group of 5 lecturers: number of friends of each members of the group: 1,
2, 3, 3 and 4
• Mean: (1+2+3+3+4)/5 = 2.6 friends per person
• Clearly an hypothetical value

• How can we know that it is an accurate model?

• Difference between the real data and the model created
 Measures of dispersion
• Calculate the magnitude of the differences between each data and the mean:

• Total error = sum of differences From Field, 2000

) = (-1.6)+(-0.6)+(0.4)+(1.4) = 0

No errors !
• Positive and negative: they cancel each other out.
 Sum of Squared errors (SS)
• To avoid the problem of the direction of the errors: we square them
• Instead of sum of errors: sum of squared errors (SS):

= (1.6) 2 + (-0.6)2 + (0.4)2 +(0.4)2 + (1.4)2

= 2.56 + 0.36 + 0.16 + 0.16 +1.96
= 5.20
• SS gives a good measure of the accuracy of the model
• But: dependent upon the amount of data: the more data, the higher the SS.
• Solution: to divide the SS by the number of observations (N)
• As we are interested in measuring the error in the sample to estimate the one in the population we
divide the SS by N-1 instead of N and we get the variance (S2) = SS/N-1
 Variance and standard deviation

• Problem with variance: measure in squared units

• For more convenience, the square root of the variance is taken to obtain a measure in
the same unit as the original measure:
• the standard deviation
• S.D. = √(SS/N-1) = √(s2) = s

• The standard deviation is a measure of how well the mean represents the data.
 Standard deviation

Small S.D.: Large S.D.:

data close to the mean: data distant from the mean:
mean is a good fit of the data mean is not an accurate representation
 SD and SEM (SEM = SD/√N)

• What are they about?

• The SD quantifies how much the values vary from one another: scatter or spread
• The SD does not change predictably as you acquire more data.

• The SEM quantifies how accurately you know the true mean of the population.
• Why? Because it takes into account: SD + sample size

• The SEM gets smaller as your sample gets larger

• Why? Because the mean of a large sample is likely to be closer to the true mean than is the
mean of a small sample.
The SEM and the sample size
A population
The SEM and the sample size
Small samples (n=3)

Sample means
Big samples (n=30)
‘Infinite’ number of samples
Samples means = 

Sample means
 SD and SEM

The SD quantifies the scatter of the data. The SEM quantifies the distribution
of the sample means.
 SD or SEM ?

• If the scatter is caused by biological variability, it is important to show the

variation.
• Report the SD rather than the SEM.
• Better even: show a graph of all data points.

• If you are using an in vitro system with no biological variability, the scatter is
about experimental imprecision (no biological meaning).
• Report the SEM to show how well you have determined the mean.
 Confidence interval
• Range of values that we can be 95% confident contains the true mean of the population.
- So limits of 95% CI: [Mean - 1.96 SEM; Mean + 1.96 SEM] (SEM = SD/√N)

Error bars Type Description

Standard deviation Descriptive Typical or average difference
between the data points and their
mean.

Standard error Inferential A measure of how variable the

mean will be, if you repeat the
whole study many times.

Confidence interval Inferential A range of values you can be 95%

usually 95% CI confident contains the true mean.
 Analysis of Quantitative Data

• Choose the correct statistical test to answer your question:

• They are 2 types of statistical tests:

• Parametric tests with 4 assumptions to be met by the data,

• Non-parametric tests with no or few assumptions (e.g. Mann-Whitney test)

and/or for qualitative data (e.g. Fisher’s exact and χ2 tests).
 Assumptions of Parametric Data
• All parametric tests have 4 basic assumptions that must be met for the
test to be accurate.
1) Normally distributed data
• Normal shape, bell shape, Gaussian shape

• Transformations can be made to make data suitable for parametric analysis.

 Assumptions of Parametric Data
• Frequent departures from normality:
• Skewness: lack of symmetry of a distribution

Skewness < 0 Skewness = 0 Skewness > 0

• Kurtosis: measure of the degree of ‘peakedness’ in the distribution

• The two distributions below have the same variance approximately
the same skew, but differ markedly in kurtosis.

More peaked distribution: kurtosis > 0 Flatter distribution: kurtosis < 0

 Assumptions of Parametric Data
2) Homogeneity in variance
• The variance should not change systematically throughout the data

3) Interval data (linearity)

• The distance between points of the scale should be equal at all parts along the scale.

4) Independence
• Data from different subjects are independent
• Values corresponding to one subject do not influence the values corresponding to another subject.
• Important in repeated measures experiments
 Analysis of Quantitative Data

• Is there a difference between my groups regarding the variable I am measuring?

• e.g. are the mice in the group A heavier than those in group B?
• Tests with 2 groups:
• Parametric: Student’s t-test
• Non parametric: Mann-Whitney/Wilcoxon rank sum test
• Tests with more than 2 groups:
• Parametric: Analysis of variance (one-way ANOVA)
• Non parametric: Kruskal Wallis

• Is there a relationship between my 2 (continuous) variables?

• e.g. is there a relationship between the daily intake in calories and an increase in body weight?
• Test: Correlation (parametric) and curve fitting
Sample Statistical inference Population

Difference Meaningful? Yes

Real?

Statistical test

Big enough? Statistic

e.g. t, F …
=

Difference + Noise + Sample

 Signal-to-noise ratio
• Stats are all about understanding and controlling variation.
Difference
Difference + Noise
Noise

signal If the noise is low then the signal is detectable …

noise = statistical significance

signal … but if the noise (i.e. interindividual variation) is large

then the same signal will not be detected
noise = no statistical significance

• In a statistical test, the ratio of signal to noise determines the significance.

 Comparison between 2 groups:
Student’s t-test
• Basic idea:
• When we are looking at the differences between scores for 2 groups, we have to judge
the difference between their means relative to the spread or variability of their scores.
• Eg: comparison of 2 groups: control and treatment
Student’s t-test
Student’s t-test
 SE gap ~ 2 n=3 SE gap ~ 4.5 n=3
13 16

15

Dependent variable
Dependent variable
12
14
11 13
~ 2 x SE: p~0.05 ~ 4.5 x SE: p~0.01
10 12

11
9
10
8 9
A B A B

SE gap ~ 2 n>=10
SE gap ~ 1 n>=10
12.0
11.5

Dependent variable
11.5
Dependent variable

11.0
11.0
~ 1 x SE: p~0.05 ~ 2 x SE: p~0.01
10.5
10.5

10.0 10.0

9.5 9.5
A B A B
 CI overlap ~ 1 n=3 CI overlap ~ 0.5 n=3

14

Dependent variable

Dependent variable
12 15

10 ~ 1 x CI: p~0.05
~ 0.5 x CI: p~0.01
8 10

A B
A B
CI overlap ~ 0.5 n>=10
CI overlap ~ 0 n>=10
12
12
Dependent variable

Dependent variable
11 11
~ 0.5 x CI: p~0.05
~ 0 x CI: p~0.01
10 10

9 9
A B A B
 Student’s t-test
• 3 types:

• Independent t-test
• compares means for two independent groups of cases.

• Paired t-test
• looks at the difference between two variables for a single group:
• the second ‘sample’ of values comes from the same subjects (mouse, petri dish …).

• One-Sample t-test
• tests whether the mean of a single variable differs from a specified constant (often 0)
Example: coyotes.xlsx

• Question: do male and female coyotes differ in size?

• Sample size
• Data exploration
• Check the assumptions for parametric test
• Statistical analysis: Independent t-test
Power analysis
• Example case:
No data from a pilot study but we have found some information in the
literature.
In a study run in similar conditions as in the one we intend to run, male coyotes
were found to measure: 92cm+/- 7cm (SD).
We expect a 5% difference between genders.
• smallest biologically meaningful difference
G*Power

Independent t-test

A priori Power analysis

Example case:

You don’t have data from a pilot study but you

have found some information in the literature.

In a study run in similar conditions to the one you

intend to run, male coyotes were found to
measure:
92cm+/- 7cm (SD)

You expect a 5% difference between genders with

a similar variability in the female sample.

You need a sample size of n=76 (2*38)

Power Analysis
 Power Analysis

H0 H1
 Power Analysis
For a range of sample sizes:
Data exploration plotting data
 Coyote
110
Maximum

100

Upper Quartile (Q3) 75th percentile

Length (cm)

90 Interquartile Range (IQR)

Median Lower Quartile (Q1) 25th percentile

80

Smallest data value Cutoff = Q1 – 1.5*IQR

> lower cutoff
70
Outlier

60
Male Female
 Assumptions for parametric tests
Histogram of Coyote (Bin size 2)
10
Females
8 Males

6
Counts

0 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98100102104106 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98100102104106

Histogram of Coyote (Bin size 3)

12
Females
10 Males
8

Normality 
Counts

0 69 72 75 78 81 84 87 90 93 96 99 102 105 69 72 75 78 81 84 87 90 93 96 99 102 105

Histogram of Coyote (Bin size 4)

14
Females
12 Males
10
Counts

0 68 72 76 80 84 88 92 96 100 104 108 68 72 76 80 84 88 92 96 100 104 108

Coyotes
110

100

90

Length (cm)
80

70

60
Females Males
Independent t-test: results

Males tend to be longer than females

but not significantly so (p=0.1045)

Homogeneity in variance 

What about the power of the analysis?

 Power analysis
You would need a sample 3 times bigger to reach the accepted power of 80%.

But is a 2.3 cm difference between genders biologically relevant (<3%) ?

 The sample size: the bigger the better?

• It takes huge samples to detect tiny differences but tiny samples to detect huge
differences.

• What if the tiny difference is meaningless?

• Beware of overpower
• Nothing wrong with the stats: it is all about
interpretation of the results of the test.

• Remember the important first step of power

analysis
• What is the effect size of biological interest?
Another example of t-test:
working memory.xlsx

A group of rhesus monkeys (n=15) performs a task involving memory after having received
a placebo. Their performance is graded on a scale from 0 to 100. They are then asked to
perform the same task after having received a dopamine depleting agent.

Is there an effect of treatment on the monkeys' performance?

Another example of t-test:
working memory.xlsx

Normality 
Another example of t-test:
working memory.xlsx
Paired t-test: Results
working memory.xlsx
0

-2

Difference in performance
-4

-6

-8

-10

-12

-14

-16

-18
 Comparison of more than 2 means
• Running multiple tests on the same data increases the familywise error rate.
• What is the familywise error rate?
– The error rate across tests conducted on the same experimental data.

• One of the basic rules (‘laws’) of probability:

– The Multiplicative Rule: The probability of the joint occurrence of 2 or more
independent events is the product of the individual probabilities.
 Familywise error rate
• Example: All pairwise comparisons between 3 groups A, B and C:
– A-B, A-C and B-C
• Probability of making the Type I Error: 5%
– The probability of not making the Type I Error is 95% (=1 – 0.05)
• Multiplicative Rule:
– Overall probability of no Type I errors is: 0.95 * 0.95 * 0.95 = 0.857

• So the probability of making at least one Type I Error is 1-0.857 = 0.143 or 14.3%
• The probability has increased from 5% to 14.3%

• Comparisons between 5 groups instead of 3, the familywise error rate is 40% (=1-(0.95)n)
 Familywise error rate
• Solution to the increase of familywise error rate: correction for multiple comparisons
– Post-hoc tests
• Many different ways to correct for multiple comparisons:
– Different statisticians have designed corrections addressing different issues
• e.g. unbalanced design, heterogeneity of variance, liberal vs conservative

• However, they all have one thing in common:

– the more tests, the higher the familywise error rate: the more stringent the correction
• Tukey, Bonferroni, Sidak, Benjamini-Hochberg …
– Two ways to address the multiple testing problem
• Familywise Error Rate (FWER) vs. False Discovery Rate (FDR)
 Multiple testing problem
• FWER: Bonferroni: αadjust = 0.05/n comparisons e.g. 3 comparisons: 0.05/3=0.016
– Problem: very conservative leading to loss of power (lots of false negative)
– 10 comparisons: threshold for significance: 0.05/10: 0.005
– Pairwise comparisons across 20.000 genes 

• FDR: Benjamini-Hochberg: the procedure controls the expected proportion of

“discoveries” (significant tests) that are false (false positive).
– Less stringent control of Type I Error than FWER procedures which control the probability of at least
one Type I Error
– More power at the cost of increased numbers of Type I Errors.

• Difference between FWER and FDR:

– a p-value of 0.05 implies that 5% of all tests will result in false positives.
– a FDR adjusted p-value (or q-value) of 0.05 implies that 5% of significant tests will result in false
positives.
 Analysis of variance
• Extension of the 2 groups comparison of a t-test but with a slightly different logic:

• t-test = mean1 – mean2

Pooled SEM Pooled SEM

• ANOVA = variance between means

Pooled SEM

Pooled SEM

• ANOVA compares variances:

– If variance between the several means > variance within the groups (random error) then the means
must be more spread out than it would have been by chance.
 Analysis of variance
• The statistic for ANOVA is the F ratio.

Variance between the groups

• F=
Variance within the groups (individual variability)

Variation explained by the model (= systematic)

• F=
Variation explained by unsystematic factors (= random variation)

• If the variance amongst sample means is greater than the error/random variance, then
F>1
– In an ANOVA, we test whether F is significantly higher than 1 or not.
 Analysis of variance
Source of variation Sum of Squares df Mean Square F p-value

Between Groups 2.665 4 0.6663 8.423 <0.0001

Within Groups 5.775 73 0.0791
In Power Analysis:
Total 8.44 77
Pooled SD=MS(Residual)

• Variance (= SS / N-1) is the mean square

– df: degree of freedom with df = N-1

Between groups variability

Within groups variability

Total sum of squares
 Example: protein.expression.csv

• Question: is there a difference in protein expression between

the 5 cell lines?

• 1 Plot the data

• 2 Check the assumptions for parametric test
• 3 Statistical analysis: ANOVA
 Protein expression
Protein expression
10

0
2
4
6
8

0
2
4
6
8
10
A

A
B

B
C

C
D

D
E

E
Parametric tests assumptions
 10

Protein expression
1

0.1
A B C D E

Transform of Protein expression

10 1.0

0.5
Protein expression

Log Protein
1 0.0

-0.5

0.1 -1.0
A B C D E A B C D E
Parametric tests assumptions
 Analysis of variance: Post hoc tests

• The ANOVA is an “omnibus” test: it tells you that there is (or not) a difference
between your means but not exactly which means are significantly different
from which other ones.

– To find out, you need to apply post hoc tests.

– These post hoc tests should only be used when the ANOVA finds a significant
effect.
Analysis of variancec
 Analysis of variance: results
Homogeneity of variance 

F=0.6727/0.08278=8.13
 Correlation
• A correlation coefficient is an index number that measures:
– The magnitude and the direction of the relation between 2 variables
– It is designed to range in value between -1 and +1
 Correlation
• Most widely-used correlation coefficient:
– Pearson product-moment correlation coefficient “r”

• The 2 variables do not have to be measured in the same units but they have to be proportional
(meaning linearly related)
– Coefficient of determination:
• r is the correlation between X and Y
• r2 is the coefficient of determination:
– It gives you the proportion of variance in Y that can be explained by X, in
percentage.
 Correlation
Example: roe deer.xlsx
• Is there a relationship between parasite burden and body mass in roe deer?

30
Male
Female
25

Body Mass
20

15

10
1.0 1.5 2.0 2.5 3.0 3.5
Parasites burden
 Correlation
Example: roe deer.xlsx

There is a negative correlation between parasite load and

fitness but this relationship is only significant for the
males(p=0.0049 vs. females: p=0.2940).
 Curve fitting
• Dose-response curves
– Nonlinear regression
– Dose-response experiments typically use around 5-10 doses of agonist, equally spaced on a
logarithmic scale
– Y values are responses

• The aim is often to determine the IC50 or the EC50

– IC50 (I=Inhibition): concentration of an agonist that provokes a response half way between the
maximal (Top) response and the maximally inhibited (Bottom) response.
– EC50 (E=Effective): concentration that gives half-maximal response

Stimulation: Inhibition:
Y=Bottom + (Top-Bottom)/(1+10^((LogEC50-X)*HillSlope)) Y=Bottom + (Top-Bottom)/(1+10^((X-LogIC50)))
 Curve fitting
Example: Inhibition data.xlsx 500
No inhibitor
400 Inhibitor
300

200

100

0
-10 -8 -6 -4 -2

Step by step analysis and considerations:

-100 log(Agonist], M

1- Choose a Model:
not necessary to normalise
should choose it when values defining 0 and 100 are precise
variable slope better if plenty of data points (variable slope or 4 parameters)
2- Choose a Method: outliers, fitting method, weighting method and replicates

3- Compare different conditions:

Diff in parameters
Diff between conditions for one or more parameters
Constraint vs no constraint
Diff between conditions for one or more parameters
4- Constrain:
depends on your experiment
depends if your data don’t define the top or the bottom of the curve
 Curve fitting
Example: Inhibition data.xlsx 500
No inhibitor
400 Inhibitor
300

200

100

0
-10 -8 -6 -4 -2
-100 log(Agonist], M

Step by step analysis and considerations:

5- Initial values:
defaults usually OK unless the fit looks funny
6- Range:
defaults usually OK unless you are not interested in the x-variable full range (ie time)
7- Output:
summary table presents same results in a … summarized way.
8 – Confidence: calculate and plot confidence intervals

9- Diagnostics:
check for normality (weights) and outliers (but keep them in the analysis)
check Replicates test
residual plots
 Curve fitting
Example: Inhibition data.xlsx
Non- normalized data 4 parameters Non- normalized data 3 parameters
500
500

450
450

400
400

350
350

300
300

250

Response
250

Response
EC50 200 EC50
200

150
150 No inhibitor
No inhibitor
100
100 Inhibitor
Inhibitor
50
50

0
0
-9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0
-9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0
-50 log(Agonist)
-50 log(Agonist)
-100
-100

Normalized data 3 parameters

Normalized data 4 parameters 110
110
100
100
90
90
80
80
70
70
Response (%)

60
60
EC50 50
50
No inhibitor 40
40
Inhibitor 30
30 No inhibitor
20 20
Inhibitor
10 10

0 0
-10.0 -9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -10.0 -9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0
log(Agonist) log(Agonist)
 Curve fitting
Example: Inhibition data.xlsx
500
Non- normalized data 4 parameters No inhibitor Inhibitor
450

No inhibitor Inhibitor 400

350

300

Replicates test for lack of fit 250

Response
SD replicates 22.71 25.52 200 EC50

SD lack of fit 41.84 32.38 150 No inhibitor

Inhibitor
Discrepancy (F) 3.393 1.610
100

P value 0.0247 0.1989

50
-7.158 -6.011
0
-9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0
Evidence of inadequate model? Yes No -50 log(Agonist)
-100

Non- normalized data 3 parameters

500

450

400
Replicates test for lack of fit 350

SD replicates 22.71 25.52 300

SD lack of fit 39.22 30.61 250

Response
Discrepancy (F) 2.982 1.438 200 EC50

P value 0.0334 0.2478 150

No inhibitor
Evidence of inadequate model? Yes No
100

50
Inhibitor -7.159 -6.017
0
-9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0
-50 log(Agonist)
-100

Normalized data 4 parameters

110

100

Replicates test for lack of fit 90

SD replicates 5.755 7.100 80

SD lack of fit 11.00 8.379

70

Response (%)
60

Discrepancy (F) 3.656 1.393 50

EC50

No inhibitor
P value 0.0125 0.2618 40
Inhibitor
Evidence of inadequate model? Yes No
30

20

-7.017 -5.943
10

0
-10.0 -9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0
log(Agonist)

Normalized data 3 parameters

110

100

90

Replicates test for lack of fit 80

SD replicates 5.755 7.100 70

SD lack of fit 12.28 9.649

60

50

Discrepancy (F) 4.553 1.847 40

P value 0.0036 0.1246 30

20
No inhibitor
Inhibitor
Evidence of inadequate model? Yes No 10
-7.031 -5.956
0
-10.0 -9.5 -9.0 -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0
log(Agonist)
 My email address if you need some help with GraphPad:

anne.segonds-pichon@babraham.ac.uk

Slides and manual available on:

https://www.bioinformatics.babraham.ac.uk/training.html