1. Power in QTL linkage analysis
Shaun Purcell & Pak Sham
SGDP, IoP, London, UK
F:\pshaun\power.ppt

2. Power primer
Statistics (e.g. chi-squared, z-score) are continuous measures of support for a certain hypothesis
YES NO
Test statistic
YES OR NO decision-making : significance testing
Inevitably leads to two types of mistake :
false positive (YES instead of NO) (Type I)
false negative (NO instead of YES) (Type II)

3. Hypothesis testing Null hypothesis : no effect
A ‘significant’ result means that we can reject the null hypothesis
A ‘nonsignificant’ result means that we cannot reject the null hypothesis

4. Statistical significance
The ‘p-value’
The probability of a false positive error if the null were in fact true
Typically, we are willing to incorrectly reject the null 5% or 1% of the time (Type I error)

5. Misunderstandings p - VALUES
that the p value is the probability of the null hypothesis being true
that high p values mean large and important effects
NULL HYPOTHESIS
that nonrejection of the null implies its truth

6. Limitations IF A RESULT IS SIGNIFICANT
leads to the conclusion that the null is false
BUT, this may be trivial
IF A RESULT IS NONSIGNIFICANT
leads only to the conclusion that it cannot be concluded that the null is false

7. Alternate hypothesis Neyman & Pearson (1928) ALTERNATE HYPOTHESIS
specifies a precise, non-null state of affairs with associated risk of error

8. P(T) Critical value T Sampling distribution if H0 were true
Sampling distribution if HA were true
P(T)
Critical value   T

9. Nonsignificant result
STATISTICS
Rejection of H0
Nonrejection of H0
Type I error
at rate 
Nonsignificant result
H0 true
R E A L I T Y
Type II error
at rate 
Significant result
HA true
POWER =(1- )

10. Power The probability of rejection of a false null-hypothesis
depends on
- the significance crtierion ()
- the sample size (N)
- the effect size (NCP)
“The probability of detecting a given effect size
in a population from a sample of size N,
using significance criterion ”

11. Impact of  alpha
P(T)
Critical value T  

12. Impact of  effect size, N
P(T)
Critical value T  

13. Applications EXPERIMENTAL DESIGN MAGNITUDE VS. SIGNIFICANCE
- avoiding false positives vs. dealing with false negatives
MAGNITUDE VS. SIGNIFICANCE
- highly significant  very important
INTERPRETING NONSIGIFICANT RESULTS
- nonsignficant results only meaningful if power is high
POWER SURVEYS / META-ANALYSES
- low power undermines the confidence that can be placed in statistically significant results

14. Practical Exercise 1
Calculation of power for simple case-control association study.
DATA : allele frequency of “A” allele for cases and controls
TEST : 2-by-2 contingency table : chi-squared
(1 degree of freedom)

15. Step 1 : determine expected chi-squared
Hypothetical allele frequencies
Cases P(A) = 0.68
Controls P(A) = 0.54
Sample 150 cases, 150 controls
Excel spreadsheet : faculty drive:\pshaun\chisq.xls
Chi-squared statistic = 12.36

16. P(T) Critical value T Step 2. Determine the critical value for a given
type I error rate, 
- inverse central chi-squared
distribution
P(T)
Critical value  T

17.
df = 1 , NCP = 0
 X
0.05
0.01
0.001

18. Step 3. Determine the power for a given critical value
and non-centrality parameter
- non-central chi-squared distribution
P(T)
Critical value T

19. Determining power df = 1 , NCP = 12.36  X Power 0.05 3.84146
0.94
0.83
0.59

20. Exercises
Using the spreadsheet and the chi-squared calculator, what is power (for the 3 levels of alpha)
1. … if the sample size were 300 for each group?
2. … if allele frequencies were 0.24 and 0.18 for 750 cases and 750 controls?

21. Answers
1. NCP =  Power
2. NCP =  Power
nb. Stata : di 1-nchi(df,NCP,invchi(df,))

22. QTL linkage POWER Type I errors Variance explained Type II errors
Sample N
Effect Size
Allele frequencies
Genetic values
Variance explained

23. Power of tests
For chi-squared tests on large samples, power is determined by non-centrality parameter () and degrees of freedom (df)
 = E(2lnL1 - 2lnL0)
= E(2lnL1 ) - E(2lnL0)
where expectations are taken at asymptotic values of maximum likelihood estimates (MLE) under an assumed true model

24. Linkage test HA H0
for i=j
for ij
for i=j
for ij

25. Expected log likelihood under H0
Expectation of the quadratic product is simply s, the sibship size
(note: standarised trait)

26. Expected log likelihood under HA

27. Linkage test Expected NCP
For sib-pairs under complete marker information
Determinant of 2-by-2 standardised covariance matrix = 1 - r2

28. Approximation of NCP NCP per sib pair is proportional to
- the # of pairs in the sibship
(large sibships are powerful)
- the square of the additive QTL variance
(decreases rapidly for QTL of v. small effect)
- the sibling correlation
(structure of residual variance is important)

29. QTL linkage POWER Type I errors Variance explained Type II errors
Allele frequencies
Genetic values
Type I errors
Type II errors
Sample N
Effect Size
Variance explained
Marker vs functional
variant
Recombination
fraction

30. Incomplete linkage
The previous calculations assumed analysis was performed at the QTL.
- imagine that the test locus is not the QTL
but is linked to it.
Calculate sib-pair IBD distribution at the QTL, conditional on IBD at test locus,
- a function of recombination fraction

31.  at QTL
1/2 1
 at M
1/2 1

32. Use conditional probabilities to calculate the sib correlation conditional on IBD sharing at the test marker. For example : for IBD 0 at marker :
 at QTL
1/2 1 r VS
VA / 2 + VS
VA + VD + VS
P(M=0 | QTL) VS
VA / 2 + VS +
C0 =
VA + VD + VS +

33. The noncentrality parameter per sib pair is then given by

34. If the QTL is additive, then
attenuation of the NCP is by a factor of (1-2)4
= square of the correlation
between the proportions of alleles IBD
at two loci with recombination fraction 

35. Effect of incomplete linkage

36. Effect of incomplete linkage

37. Comparison to H-E Amos & Elston (1989) H-E regression
- 90% power (at significant level 0.05)
- QTL variance 0.5
- marker and major gene are completely linked
 320 sib pairs
 778 sib pairs if  = 0.1

38. GPC input parameters Proportions of variance additive QTL variance
dominance QTL variance
residual variance (shared / nonshared)
Recombination fraction ( )
Sample size & Sibship size ( )
Type I error rate
Type II error rate

39. GPC output parameters Expected sibling correlations
- by IBD status at the QTL
- by IBD status at the marker
Expected NCP per sibship
Power
- at different levels of alpha given sample size
Sample size
- for specified power at different levels of alpha given power

40. From GPC Modelling additive effects only Sibships Individuals
Pairs (320) 530
Pairs ( = 0.1) 666 (778) 1332
Trios ( = 0.1)
Quads ( = 0.1)
Quints ( = 0.1)

41. Practical Exercise 2
What is the effect on power to detect linkage of :
1. QTL variance?
2. residual sibling correlation?
3. marker-QTL recombination fraction?

42. Pairs required (=0, p=0.05, power=0.8)

43. Pairs required (=0, p=0.05, power=0.8)

44. Effect of residual correlation
QTL additive effects account for 10% trait variance
Sample size required for 80% power (=0.05)
No dominance
 = 0.1
A residual correlation 0.35
B residual correlation 0.50
C residual correlation 0.65

45. Individuals required

46. Selective genotyping Unselected Proband Selection EDAC
Maximally Dissimilar
ASP
Extreme Discordant
EDAC
Mahanalobis Distance

47. Selective genotyping
The power calculations so far assume an unselected population.
- calculate expected NCP per sibship
If we have a sample with trait scores
- calculate expected NCP for each sibship conditional on trait values
- this quantity can be used to rank order the sample for genotying

48. Sibship informativeness : sib pairs-4 -3 -2 -1 1 2 3 4
Sib 1 trait
Sib 2 trait
0.2
0.4
0.6
0.8
1.2
1.4
1.6
Sibship NCP

49. Sibship informativeness : sib pairs-4 -3 -2 -1 1 2 3 4
Sib 1 trait
Sib 2 trait
0.5
1.5
Sibship NCP -4 -3 -2 -1 1 2 3 4
Sib 1 trait
Sib 2 trait
0.5
1.5
Sibship NCP
dominance -4 -3 -2 -1 1 2 3 4
Sib 1 trait
Sib 2 trait
0.5
1.5
Sibship NCP
unequal allele
frequencies
rare
recessive

50. Selective genotyping SEL T ASP PS ED EDAC MaxD MDis SEL B p d/a .5
15.82 .1
17.10
.25
15.45 .1 1
16.88
.25 1
15.76 .5 1
18.89
.75 1
27.64 .9
43.16 1

51. Impact of selection

52. QTL linkage POWER Type I errors Variance explained Type II errors
Allele frequencies
Genetic values
Type I errors
Type II errors
Sample N
Effect Size
Variance explained
Marker vs functional
variant
Recombination
fraction
Locus
informativeness
PIC

53. Indices of marker informativeness:
Markers should be highly polymorphic
- alleles inherited from different sources are likely to be distinguishable
Heterozygosity (H)
Polymorphism Information Content (PIC)
- measure number and frequency of alleles at a locus

54. Heterozygosity n = number of alleles,
pi = frequency of the ith allele.
H = probability that an individual is heterozygous

55. Heterozygosity Allele Frequency Genotype Frequency 11 0.04 12 0.14
Genotype Frequency 11 22 33 44
H = 0.675

56. Polymorphism information content
IF a parent is heterozygous,
their gametes will usually be informative.
BUT if both parents & child are heterozygous for the same genotype,
origins of child’s alleles are ambiguous
IF C = the probability of this occurring,
PIC = H - C

57. Polymorphism information content

58. Possible IBD configurations given parental genotypes
Parental Mating
Type
Configuration
Probability
1 Hom  Hom 1/ / (1-H)2
2 Hom  Het / H(1-H)
3 Hom  Het / / H(1-H)
1: AA AA / AA AA
2: AA AB / AA AB
3: AA AB / AA AA
4: AB AB / AA AB
5: AB AB / AA BB
6: AB AB / AA AA
7: AB AB / AB AB
Het  Het / H2 / 2
5 Het  Het (H2 -C)/4
6 Het  Het (H2 -C)/4
7 Het  Het / / C/2

59. PIC & NCP for linkage
From the table of possible IBD configurations given parental genotypes,
Therefore, NCP is attenuated in proportion to PIC

60. QTL linkage POWER Type I errors Variance explained Type II errors
Allele frequencies
Genetic values
Type I errors
Type II errors
Sample N
Effect Size
Variance explained
Marker vs functional
variant
Recombination
fraction
Locus
informativeness
PIC
Multipoint
Marker density
MPIC

61. Multipoint IBD
Estimates IBD sharing at any arbitrary point along a chromosomal region, using all available marker information on a chromosome simultaneously.

62. , ,  and PIC ^

63. 2. Convert map distances into recombination fractions
1. Calculate PIC for each marker
PIC
0.41
0.77
0.84
0.79
5 cM -->  =
2. Convert map distances into recombination fractions
Haldane map function (m = map distance in Morgans)
5cM
5cM
5cM
5cM M1
0.1
0.2
0.7 M2
0.2 M3 M4
0.2
0.1 M5
0.2

64. 3. Calculate covariance matrix between pi-hat at markers
M1 M2 M3 M4 M5 M M M M M
MM =

65. 4. Consider each multipoint position
5cM M1 M2 M3 M4 M5
10cM
15cM
20cM
25cM
30cM
At each position along the chromosome, calculate covariance between trait locus and each of the markers
0.0344
0.0528
0.0472
0.0363
0.0290
MT = MD 10 15 20 25 30 RF
0.091
0.130
0.165
0.197
0.226
PIC
0.41
0.77
0.84
0.79

66. Fulker et al multipoint
If is a vector of single marker IBD estimates then a multipoint IBD estimate at test position t is given by :
Conditional on the variance of  at the test position is reduced by a quantity which can be thought of as a multipoint PIC

67. 5. Calculate MPIC

68. 10 cM map

69. 5 cM map

70. Exclusion mapping
Exclusion : support for the hypothesis that a QTL of at least a certain effect is absent at that position
Normally, the LRT compares the likelihood at the MLE and the null
In exclusion mapping, the LRT compares the likelihood of a fixed effect size against the null and therefore can be negative

71. Conclusions Factors influencing power QTL variance Sib correlation
Sibship size
Marker informativeness
Marker density
Phenotypic selection