![Page 1: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/1.jpg)
22 October 2014
Analysing qPCR outcomesLecture Analysis of Variance by Dr Maartje Klapwijk
![Page 2: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/2.jpg)
01
Personal BackgroundSince 2009 Insect Ecologist at SLU Climate Change and other anthropogenic effects on interaction between predator and preyMostly field based research
TeachingBasic Statistics and ANOVA during PhDExperimental design and ANOVA at SLU
![Page 3: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/3.jpg)
Program
✤ Descriptive statistics
✤ t - test
✤ one-way ANOVA
✤ General Linear Model
![Page 4: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/4.jpg)
01
VariabilitySpread of data points in the data set
Anova measures and compares variability within and between groups
http://humanorigins.si.edu/evidence/genetics/skin-color
![Page 5: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/5.jpg)
Statistics to describe data
✤ Mean
✤ Variance
✤ Standard deviation
✤ Standard Error of the Mean
✤ Confidence intervals
![Page 6: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/6.jpg)
The mean
x = x1 + x2 + x3 + x4 .....+ xnn
VP1LEC
200
150
100
50
0
embryo
MEAN
![Page 7: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/7.jpg)
Variance
Var(x) =σ 2 = [(x − x )2 ]∑
VP1LEC
200
150
100
50
0
embryo
MEAN
Var (LEC) = 4864.40
Var (VP1) = 4.006
![Page 8: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/8.jpg)
Standard Deviation
s =σ = σ 2
Var (LEC) = 4864.40
Var (VP1) = 4.006
s (LEC) = 69.7
s (VP1) = 2.001
![Page 9: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/9.jpg)
VP1LEC
200
150
100
50
0
embryo
MEAN
Interval Plot of MEANBars are One Standard Error from the Mean
Standard Error of the Mean
SEM = σn
SEM(LEC) = 69.710
= 22.1
SEM(VP1) = 2.00110
= 0.63
![Page 10: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/10.jpg)
![Page 11: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/11.jpg)
Confidence intervals
CI = x ± tα /2,d . f × SEM
Critical t value for α = 0.05 for 95% confidence interval
95% CI for LEC = 49.28-22.1*2.2;49.28+22.1*2.2[0.66; 97.9]
95% CI for VP1=
![Page 12: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/12.jpg)
Probability distributions
What is probability?The statistical number of outcomes considered divided by the number of all outcomes
![Page 13: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/13.jpg)
Simple 1-sample t-test
Question: How likely is it that a data point comes from a collection of data with a known mean and standard deviation?
![Page 14: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/14.jpg)
2-sample t-test
Question: How likely is it that two datasets come from a collection of data with a known mean and standard deviation?
Similar to the one-sample t-tests only it compares two datasets
Example:Two groups of people have been measured for height. Question: is the height in group A similar to the height in group B?
The 2-sample t-test compares the mean and variance
When would the test show that there is a difference?
![Page 15: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/15.jpg)
![Page 16: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/16.jpg)
Parametric model assumptions
✤ Independence of samples and error
✤ Normality of error
✤ Equal variances
✤ additivity of treatments
![Page 17: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/17.jpg)
Non-parametric models
When the data does not comply to the model assumptionsone cannot use parametric models
Non-parametric models use other ways of establishing differences
![Page 18: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/18.jpg)
Independence
Data-points are independent if knowing the error of one or a subset of data-points provides no knowledge of the error of any others
![Page 19: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/19.jpg)
Normality of error
Refers to the shape of the distribution of the residuals around the model
Residuals = observed values - the fitted values
![Page 20: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/20.jpg)
Residuals
= Treatment mean - observed values
![Page 21: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/21.jpg)
Equal variances
This assumption refers to the shape of the distribution of the residuals Y
X
Figure 9.1a
X
Y
Figure 9.1b
X
Y
Figure 9.1c
X
Y
Figure 9.1d
![Page 22: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/22.jpg)
Linearity/Additivity
GLM estimates a linear relationship between the response variables and the explanatory variables
y = a + bx + ε
y = x1+ x2 + ε
![Page 23: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/23.jpg)
one-way ANOVA
Question: Are my treatment groups different?H1: They are differentH0: They are not different
Again comparison of the mean and variances
![Page 24: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/24.jpg)
Total Sums of Squares
SST = y2 −y∑( )2
kn∑
The total sums of squares represent the total variability within the dataset
k=number of levels in a treatmentn=number of replicates in each level
![Page 25: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/25.jpg)
Treatment Sums of Squares
The treatment sums of squares are a representation of the variability of the data within the groups of the treatment
SSA =C 2∑n
−y∑( )2
kn
C=treatment totalthis is the sum of all n replicates within a given level
![Page 26: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/26.jpg)
Error Sums of Squares
Error = unexplained variation in the data set
SSE = SST − SSA
![Page 27: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/27.jpg)
Our data
We have done experiment where we grew a crop on 3 different types of soil
![Page 28: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/28.jpg)
Before the analysis (http://www.slideshare.net/lssblackbelt/test-for-equal-variances)
![Page 29: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/29.jpg)
Interval plot for yield vs soil
![Page 30: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/30.jpg)
oneway ANOVA
![Page 31: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/31.jpg)
Degrees of Freedom = n - 1
![Page 32: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/32.jpg)
Sums of Squares
SSA =C 2∑n
−y∑( )2
kn
SST = y2 −y∑( )2
kn∑
SST
SSA
SSE
Partitioning of varianceSST = sum((yield-mean(yield))2) = 414.7
![Page 33: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/33.jpg)
Error = (sum((sand-mean(sand))2))+(sum((loam-mean(loam))2)) +(sum((clay-mean(clay))2)) = 315.5
Thus follows:SST(414.7)-SSE(315.5)=SSA(99.2)
Within group variability Between group variability
![Page 34: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/34.jpg)
Mean sum of squares = weighted SS for degrees of freedom
MSsoil = 99.2/2= 49.6
MSerror = 315.5/27 = 11.7
![Page 35: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/35.jpg)
Mean sum of squares = weighted SS for degrees of freedom
MSsoil = 99.2/2= 49.6
MSerror = 315.5/27 = 11.7
MSA/df = F = 49.6/11.7 = 4.24
![Page 36: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/36.jpg)
F -distribution
![Page 37: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/37.jpg)
Diagnostics
![Page 38: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/38.jpg)
![Page 39: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/39.jpg)
Where are the differences?
![Page 40: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/40.jpg)
Tukey test for differencesGrouping Information Using Tukey Method
Soil N Mean Grouping Loam 10 14,300 A Clay 10 11,500 A B Sand 10 9,900 B
Means that do not share a letter are significantly different.
Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Soil
Individual confidence level = 98,04%
Soil = Clay subtracted from:
Soil Lower Center Upper ------+---------+---------+---------+--- Loam -0,994 2,800 6,594 (-------*------) Sand -5,394 -1,600 2,194 (-------*------) ------+---------+---------+---------+--- -5,0 0,0 5,0 10,0
Soil = Loam subtracted from:
Soil Lower Center Upper ------+---------+---------+---------+--- Sand -8,194 -4,400 -0,606 (------*-------) ------+---------+---------+---------+--- -5,0 0,0 5,0 10,0
![Page 41: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/41.jpg)
General Linear Models
Advantages:
✤ handles unbalanced designs✤ can include continuous variables (covariates)✤ can include interactions✤ can include random effects
Assumptions and underlying rational the same as ANOVA
![Page 42: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/42.jpg)
General Linear Model (GLM)
Moving to some qPCR data
Treatment: 10𝜇M TSA added to the growth mediumControl: growth medium
Expressions are measured every week for 4 weeks
Data collected: 3 technical replicates (constitutes one biological replicate)3 biological replicates4 weeks of expressions
resulting in 72 replicates (but only 24 biological replicates)
![Page 43: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/43.jpg)
Experiment VP1 - 10𝜇M TSA
Response: Expression
Explanatory:Treatment
- control- treatment
Week (continuous)
![Page 44: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/44.jpg)
Model formula
Expression= treatment + week+treatment*week+ ε
![Page 45: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/45.jpg)
GLM output - ANOVA tableGeneral Linear Model: Expression versus Contorl
Factor Type Levels Values Ctrl fixed 2 Control; Treatment
Analysis of Variance for Expression, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P Treatment 1 0,90704 0,13409 0,13409 7,83 0,011 week 1 0,05940 0,05940 0,05940 3,47 0,077 Trtm*week 1 0,00061 0,00061 0,00061 0,04 0,852 Error 20 0,34245 0,34245 0,01712 Total 23 1,30949
S = 0,130852 R-Sq = 73,85% R-Sq(adj) = 69,93%
![Page 46: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/46.jpg)
GLM output - coefficients
Term Coef SE Coef T P
Constant 0,37669 0,06543 5,76 0,000
Treatment Control -0,18309 0,06543 -2,80 0,011
week -0,04450 0,02389 -1,86 0,077
week*Trtm Control -0,00453 0,02389 -0,19 0,852
![Page 47: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/47.jpg)
Results:
Only treatment influences gene
expression
Mean of the control is -0.18 lower than the overall mean.
Mean of treatment is 0.18 higher than the overall mean
![Page 48: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/48.jpg)
GLM - week = category
![Page 49: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/49.jpg)
GLM - week = category
![Page 50: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/50.jpg)
GLM - week = category
![Page 51: Analysing qPCR outcomes - SLU.SE€¦ · Analysing qPCR outcomes Lecture Analysis of Variance by Dr Maartje Klapwijk. 01 Personal ... The statistical number of outcomes considered](https://reader034.vdocuments.site/reader034/viewer/2022043012/5fa81553a3dafc3c915a62e5/html5/thumbnails/51.jpg)
Some more general points
✤ Handling technical replicates
✤ Sample size estimation
✤ Ethics of data handling