Formulas for Map Distance, Interference and Significance Analysis From Tetrad Data

P = The number of Parental Ditypes (All four spores have parental marker arrangement)

N = The number of NonParental Ditypes (All four spores have nonparental marker arrangement)

T = The number of Tetratypes (Two spores are parental and two spores are nonparental)

Sample Size (n)

n = T + P + N

fP = P/n

fN = N/n

fT = T/n

Sampling Distribution of (fT, fN, fP)

Var[fT] = (fT)(1 - fT)/n

Var[fN] = (fN)(1 - fN)/n

Cov[fT,fN] = -(fT)(fN)/n

The Perkins Equation

Perkins 1949 Genetics 34:607

Morgans = X = (fT/2) + (3fN)

cM = 100X = (100 (6N + T))/(2(P +N+T))

Sampling Variance of the Perkins Equation

Var[X] = 0.25Var[fT] + 9Var[fN] + 3Cov[fT,fN]

The Standard Error of X

The Standard Error (SE) of X is calculated as follows.

S.E. = sqrt(Var[X])

Where sqrt(z) stands for the square root of z.

The Standard Error of a Difference in Two Map Distances

Var[X1-X2] = Var[X1] + Var[X2]

SE (X1-X2) = sqrt(Var[X1-X2])

Where X1 and X2 are two intervals in Morgans.

Evaluating the Significance of Differences in Map Distances From the Perkins Equation

From Russ Lande

To compare two map distances determined by the Perkins Equation to see if they are significantly different.

Method 1:

Calculate X, Var[X] and SE for the two intervals (X1 and X2). If the error bars around the individual X values don't overlap, then the differences may be real.

Does X1 +/- SE of X1 overlap X2 +/- SE of X2 ?

Method 2:

Calculate the Var[X1-X2] and take the square root to get the SE.

If the absolute value of the difference between the two map distances is greater than twice the standard error, then the difference is significant.

Is 2*sqrt( Var[X1] + Var[X2]) < |X1 - X2| ?

Interference by Papazian (1952; Genetics 37: 175-188).

If no interference, then you can calculate the number of NPDs (NonParental Ditypes) expected as:

E = fraction of NPDs expected = 0.5 [ (1-fT) - (1-(3fT/2))^(2/3)]

Where ^ denotes exponent and fT is observed fraction of TTs.

(F) = fraction of NPD's observed / fraction of NPDs expected, so then

F = fN / E

An Approximation of the Sampling Variance of F

From Russ Lande.

If the Standard Error (SE) of F is much less than F, then this formula should be reasonably accurate.

Var[F] = ((F^2) / n) [( (1-fN)/ fN) - (B fT/ E ) + ( ( (B^2) fT (1-fT) )/(4 E^2) )]

Where ^ denotes exponent and B is defined as follows.

B = 1 - ( 1 - 3fT / 2)^(-1/3)

SE of F= sqrt( Var[F])

Interference by a "Better Way"

This two-factor test for interference is both more efficient than the NDP ratio test and valid for any ƒTobs and ƒNobs.  The underlying equations are all exact under the assumption of no interference.

1.  The recombination frequency (R) is calculated from ƒNobs and  ƒTobs:

R = ƒNobs + ƒTobs /2
Eq. 2

2.  X, the map length in Morgans under the assumption of no interference is
calculated as:

X = - 1/2ln(1-2R)                        Eq. 3

3.  The frequency of TTs expected is:

ƒTexp = 2/3(1-e-3X)                      Eq. 4

4.  The frequency of NPDs expected is:

ƒNexp = 1/2[(1-e-2X) - ƒTexp ]      Eq. 5

5.  The frequency of PDs expected is:

ƒPexp = 1 - ƒTexp - ƒNexp

6.  ƒPexp, ƒTexp and ƒNexp are multiplied by total tetrads to obtain the expected numbers of the three tetrad classes Pexp, Texp and Nexp.

7.  Chi-square, with one degree of freedom, is calculated as:

χ2 = (Pobs - Pexp)2/Pexp + (Tobs - Texp)2/Texp + (Nobs - Nexp)2/Nexp

where Pobs, Tobs and Nobs are the observed numbers of the three tetrad types.

Analysis of the Statistical Significance of Differences Between Two Ratios of Map Distances Determined by the Perkins Equation

The following analysis is for comparing two different ratios of map distances. Such an analysis would be useful for the following example:

Construct two strains, say A and B, with the same interval X in them, and then modify each of these strains in the same way yielding strains A' and B' each with interval X in them. This same interval X is now in four different strain backgrounds A, A', B and B'. If XA represents the map distance in Morgans of interval X in strain A, XA' represents the map distance in Morgans of interval X in strain A', XB represents the map distance in Morgans of interval X in strain B, and XB' represents the map distance in Morgans of interval X in strain B', then you can compare the ratios XA/XA' and XB/XB'.

Here are the formulas for this analysis.

Ratios

Let R1 = num1/den1

Let R2 = num2/den2

Variance of a Ratio

Var [R] = (Var [num]/(den^2)) + ( ( (num^2) (Var[den]) ) / (den^4) )

Var [R1 - R2] = Var [R1] + Var[R2]

Standard Error of a Ratio

S.E. of a Ratio = sqrt( Var[R])

Comparison of Ratios to Determine if They are Significantly Different

If the absolute value of the differences between the ratios is greater then k times the square root of the Variance of the difference between the ratios, then the ratios are significantly different from each other.

Is |R1 - R2| > k * sqrt (Var [R1] + Var[R2]) ?

Where k = 1.96 if there is no predicted direction for the difference (two tailed test) or, k = 1.65 if there is a predicted direction for the difference (one tailed test). If this equation is true, then the ratios are significantly different from each other.