ANOVA method for data analysis
View Patent ↗The present invention provides improved ANOVA methods for analyzing measured data and transformed data. The improved ANOVA method takes two data types as its input, one is the measurements, the other is a predetermined error associated with the measurements. The latter can come from a technology/platform-specific error model. Because of the additional input information, the statistical power is increased. The methods of the invention is particularly useful for analyzing gene or protein expression data.
1. A method of analyzing a plurality of measurements of a biological variable {y ti } in k different measurement groups by ANOVA analysis to determine whether there are differences among said k different measurement groups, wherein the biological variable is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis, wherein y ti is the ith measurement in the tth measurement group, t=1, 2, . . . , k and i=1, 2, . . . , n t , n t being the number of measurements in the tth measurement group, wherein each said measurement group consists of n t measurements of said biological variable under a condition common to said measurement group, and wherein each said measurement y ti has a predetermined measurement variance σ ti 2 , said method comprising
on a computer
receiving, by the computer, data representing the plurality of measurements of a biological variable {y ti } in k different measurement groups;
determining, by the computer, the predetermined measurement variance σ ti 2 , wherein said predetermined measurement variance σ ti 2 of each said measurement y ti is determined according to an error model, wherein said error model is a three-term error model according to equation
σ ti 2 =c 2 +b 2 ·y ti +a 2 ·y ti 2
wherein a is a fractional error coefficient, b is a Poisson error coefficient, and c is a standard deviation of background noise, the predetermined measurement variance being based on prior measurements of the same biological variable using the same microarray or two-dimensional gel electrophoresis;
(a) determining, by the computer, a within-group variance for said k different measurement groups, wherein said within-group variance consists of a propagated variance and a scattered variance, said propagated variance being determined based on said predetermined measurement variances {σ ti 2 } of said plurality of measurements {y ti }, and said scattered variance being determined based on deviations of said measurements in said plurality of measurements with respect to their respective group means { y t }, wherein each said group mean y t is the mean of said n t measurements in said tth measurement group, wherein said within-group variance is determined based on group variances {σ y t 2 } of said k different measurement groups, said group variance σ y t 2 for said tth measurement group consisting of a measurement group propagated variance σ y tP 2 and a measurement group scattered variance σ y tS 2 , wherein said measurement group propagated variance σ y tP 2 is determined according to the equation
σ
y
_
t
p
2
=
∑
i
=
1
n
t
σ
ti
2
n
t
2
and wherein said measurement group scattered variance σ y tS 2 is determined according to the equation
σ
y
_
t
s
2
=
1
(
n
t
-
1
)
·
∑
i
=
1
n
t
(
y
ti
-
y
_
t
)
2
;
(b) determining, by the computer, a between-group variance for said k different measurement groups, wherein said between-group variance is a variance of said group means { y t } with respect to a mean of said plurality of measurements {y ti };
(c) comparing, by the computer, said within-group variance with said between-group variance to determine a result representing whether there are any differences there between; and
(d) outputting to a user, a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying: the result of said comparing step (c) in order to provide the result of the analysis of the plurality of measurements of the biological variable, which is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis;
wherein said predetermined measurement variances {σ ti 2 } are determined prior to said steps (a)-(d); thereby determining whether there are differences among said k different measurement groups.
2. The method of claim 1 , wherein said group propagated variance and said group scattered variance are combined according to the equation
σ
y
_
t
=
σ
y
_
t
p
+
(
n
t
-
1
)
·
σ
y
_
t
s
n
t
.
3. The method of claim 2 , wherein said comparing step comprises determining the significance level of a statistical metric by a statistical test, wherein said statistical metric is determined by a method comprising
(i) determining a within-group degree of freedom;
(ii) determining a between-group degree of freedom;
(iii) determining a within-group mean square;
(iv) determining a between-group mean square; and
(v) calculating said statistical metric.
4. The method of claim 3 , wherein said within-group degree of freedom is calculated according to the equation
v
R
=
N
-
k
+
∑
t
=
1
k
1
n
t
where v R is said within-group degree of freedom, said between-group degree of freedom is calculated according to the equation
v T =k− 1
where v T is said between-group degree of freedom, said within-group mean square is calculated according to the equation
s R 2 =S R /v R
where S R 2 is said within-group mean square and where S R is calculated according to the equation
S
R
=
∑
t
=
1
k
[
v
Rt
·
(
σ
y
_
t
p
+
(
n
t
-
1
)
·
σ
y
_
t
s
)
2
n
t
]
where v Rt is calculated according to the equation
v
Rt
=
n
t
-
1
+
1
n
t
and said between-group mean square is calculated according to the equation
s T 2 =S T /v T
where s T 2 is said between-group mean square and where S T is calculated according to the equation
S
T
=
∑
t
=
1
k
n
t
·
(
y
_
t
-
y
_
)
2
where y is calculated according to the equation
y
_
=
∑
t
=
1
k
∑
i
=
1
n
t
y
t
i
N
where N is the total number of measurements.
5. The method of claim 4 , wherein said statistical test is an F-test and said significance level is a p-value determined according to the equation
p value=1 −fcdf ( s T 2 /s R 2 ,v T ,v R ).
6. The method of claim 5 , wherein said variable is the transcript level of a gene.
7. The method of claim 6 , wherein said transcript level is measured using a DNA microarray.
8. The method of claim 7 , wherein said variable is the abundance of a protein.
9. The method of claim 8 , wherein said abundance is measured using a protein microarray.
10. The method of claim 8 , wherein said abundance is measured using two-dimensional gel electrophoresis.
11. A method for analyzing variation among a plurality of measurements of a biological variable {y t } to determine whether there are differences among said plurality of measurements, wherein the biological variable is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis, wherein y t is the tth measurement, t=1, 2, . . . , n t , n t being the number of measurements, wherein said plurality of measurements are measured under a common condition, wherein each said measurement has a predetermined measurement variance σ t 2 , said method comprising
on a computer
receiving, by the computer, data representing the predetermined measurement variance σ t 2 , the predetermined measurement variance being based on prior measurements of the same biological variable,
determining, by the computer, an error-weighted measurement for each measurement in said plurality of measurements by weighting each measurement in said plurality of measurements with a weighting factor, wherein said weighting factor is determined based on said predetermined measurement variance of said measurement;
(a) determining, by the computer, a propagated variance and a scattered variance, wherein said propagated variance is determined based on said predetermined measurement variances {σ t 2 } of said plurality of measurements {y t } and wherein said scattered variance is a variance of said plurality of measurements with respect to the mean y of said plurality of measurements;
(b) comparing, by the computer, said propagated variance and said scattered variance, thereby determining whether there are differences among said plurality of measurements; and
(c) outputting to a user, a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying: a result of said comparing step (b) in order to provide the result of the analysis of the plurality of measurements of the biological variable, which is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis;
wherein said predetermined measurement variances {σ t 2 } are determined prior to said steps (a)-(c).
12. The method of claim 11 , wherein said weighting factor is determined according to the equation
w
t
i
=
1
σ
t
i
2
where σ t 2 is said predetermined measurement variance of measurement y t .
13. The method of claim 11 , wherein said comparing is carried out using an F-test according to the equation
Consistency — p value=1 −fcdf (σ y s 2 /σ y t p 2 ,n t −1, n t ).
14. A computer system comprising
a processor, and
a memory coupled to said processor and encoding one or more programs,
wherein said one or more programs cause the processor to carry out a method of analyzing a plurality of measurements of a biological variable {y ti } in k different measurement groups by ANOVA analysis to determine whether there are differences among said k different measurement groups, wherein y ti is the ith measurement in the tth measurement group, t=1, 2, . . . , k and i=1, 2, . . . , n t , n t being the number of measurements in the tth measurement group, wherein each said measurement group consists of n t measurements of said biological variable under a condition common to said measurement group, and wherein each said measurement y ti has a predetermined measurement variance σ ti 2 , said method comprising
determining the predetermined measurement variance σ ti 2 , wherein said predetermined measurement variance σ ti 2 of each said measurement y ti is determined according to an error model, wherein said error model is a three-term error model according to equation
σ ti 2 =c 2 +b 2 ·y ti +a 2 ·y ti 2
wherein a is a fractional error coefficient, b is a Poisson error coefficient, and c is a standard deviation of background noise
(a) determining a within-group variance for said k different measurement groups, wherein said within-group variance consists of a propagated variance and a scattered variance, said propagated variance being determined based on said predetermined measurement variances {σ ti 2 } of said plurality of measurements {y ti }, the predetermined measurement variances being based on prior measurements of the same biological variable using the same microarray or two-dimensional gel electrophoresis, and said scattered variance being determined based on deviations of said measurements in said plurality of measurements with respect to their respective group means { y t }, wherein each said group mean y t is the mean of said n t measurements in said tth measurement group;
(b) determining a between-group variance for said k different measurement groups, wherein said between-group variance is a variance of said group means { y t } with respect to a mean of said plurality of measurements {y ti };
(c) comparing said within-group variance with said between-group variance; and
(d) outputting to a user, a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying: a result of said comparing step (c);
wherein said predetermined measurement variances {σ ti 2 } are determined prior to said steps (a)-(d); thereby determining whether there are differences among said k different measurement groups and wherein said propagated variance is determined according to the equation
σ
y
_
t
p
2
=
∑
i
=
1
n
t
σ
t
i
2
n
t
2
and wherein said scattered variance is determined according to the equation
σ
y
_
t
s
2
=
1
(
n
t
-
1
)
·
∑
i
=
1
n
t
(
y
t
i
-
y
_
t
)
2
.
15. A computer program product for use in conjunction with a computer having a processor and a memory connected to the processor, said computer program product comprising a computer readable storage medium having a computer program mechanism encoded thereon, wherein said computer program mechanism may be loaded into the memory of said computer and cause said computer to carry out a method of analyzing a plurality of measurements of a biological variable {y ti } in k different measurement groups by ANOVA analysis to determine whether there are differences among said k different measurement groups, wherein y ti is the ith measurement in the tth measurement group, t=1, 2, . . . , k and i=1, 2, . . . , n t , n t being the number of measurements in the tth measurement group, wherein each said measurement group consists of n t measurements of said biological variable under a condition common to said measurement group, and wherein each said measurement y ti has a predetermined measurement variance σ ti 2 , said method comprising
receiving, on a computer, data representing the plurality of measurements of a biological variable {y ti } in k different measurement groups;
determining the predetermined measurement variance σ ti 2 , wherein said predetermined measurement variance σ ti 2 of each said measurement y ti is determined according to an error model based on a fractional error coefficient, a Poisson error coefficient, and a standard deviation of background noise, the predetermined measurement variance being based on prior measurements of the same biological variable using the same microarray or two-dimensional gel electrophoresis;
(a) determining a within-group variance for said k different measurement groups, wherein said within-group variance consists of a propagated variance and a scattered variance, said propagated variance being determined based on said predetermined measurement variances σ ti 2 of said plurality of measurements {y ti }, and said scattered variance being determined based on deviations of said measurements in said plurality of measurements with respect to their respective group means { y t } wherein each said group mean y t is the mean of said n t measurements in said tth measurement group, wherein said within-group variance is determined based on group variances {σ y t 2 } of said k different measurement groups, said group variance σ y t 2 for said tth measurement group consisting of a measurement group propagated variance σ y tP 2 and a measurement group scattered variance σ y tS 2 , wherein said measurement group propagated variance σ y tP 2 is determined according to the equation
σ
y
_
t
p
2
=
∑
i
=
1
n
t
σ
ti
2
n
t
2
and wherein said measurement group scattered variance σ y tS 2 is determined according to the equation
σ
y
_
t
s
2
=
1
(
n
t
-
1
)
·
∑
i
=
1
n
t
(
y
ti
-
y
_
t
)
2
;
(b) determining a between-group variance for said k different measurement groups, wherein said between-group variance is a variance of said group means { y t } with respect to a mean of said plurality of measurements {y ti };
(c) comparing said within-group variance with said between-group variance; and
(d) outputting to a user, a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying: a result of said comparing step (c) in order to provide the result of the analysis of the plurality of measurements of the biological variable, which is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis;
wherein said predetermined measurement variances σ ti 2 are determined prior to said steps (a)-(d); thereby determining whether there are differences among said k different measurement groups.
16. A computer system comprising
a processor, and
a memory coupled to said processor and encoding one or more programs,
wherein said one or more programs cause the processor to carry out a method for analyzing variation among a plurality of measurements of a biological variable {y t } to determine whether there are differences among said plurality of measurements, wherein y t is the tth measurement, t=1, 2, . . . , n t , n t being the number of measurements, wherein said plurality of measurements are measured under a common condition, wherein each said measurement has a predetermined measurement variance σ t 2 , said method comprising
(a) determining a propagated variance based on said predetermined measurement variances {σ t 2 } of said plurality of measurements {y t }, wherein said propagated variance is determined according to the equation
σ
y
_
t
p
2
=
∑
i
=
1
n
t
σ
t
2
n
t
2
;
wherein the predetermined measurement variances are based on prior measurements of the same biological variable using the same microarray or two-dimensional gel electrophoresis; and
(b) outputting to a user, a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying: a result of said determining step (a) in order to provide the result of the analysis of the plurality of measurements of the biological variable, which is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis.
17. A computer program product for use in conjunction with a computer having a processor and a memory connected to the processor, said computer program product comprising a computer readable storage medium having a computer program mechanism encoded thereon, wherein said computer program mechanism may be loaded into the memory of said computer and cause said computer to carry out a method for analyzing variation among a plurality of measurements of a biological variable {y t } to determine whether there are differences among said plurality of measurements, wherein y t is the tth measurement, t=1, 2, . . . , n t , n t being the number of measurements, wherein said plurality of measurements are measured under a common condition, wherein each said measurement has a predetermined measurement variance σ t 2 , said method comprising
(a) determining a propagated variance based on said predetermined measurement variances {σ t 2 } of said plurality of measurements {y t } wherein said propagated variance is determined according to the equation
σ
y
_
t
p
2
=
∑
i
=
1
n
t
σ
t
2
n
t
2
,
wherein the predetermined measurement variances are based on prior measurements of the same biological variable using the same microarray or two-dimensional gel electrophoresis; and
(b) outputting to a user, a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying: a result of said determining step (a) in order to provide the result of the analysis of the plurality of measurements of the biological variable, which is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis.
18. A method for analyzing variation among a plurality of measurements of a biological variable {y t } to determine whether there are differences among said plurality of measurements, wherein the biological variable is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis, wherein y t is the tth measurement, t=1, 2, . . . , n t , n t being the number of measurements, wherein said plurality of measurements are measured under a common condition, wherein each said measurement has a predetermined measurement variance σ t 2 , said method comprising
on a computer
receiving, by the computer, data representing the plurality of measurements of a biological variable {y t };
determining, by the computer, the predetermined measurement variance σ t 2 , wherein said predetermined measurement variance σ t 2 of each said measurement y t is determined according to an error model based on a fractional error coefficient, a Poisson error coefficient, and a standard deviation of background noise, the predetermined measurement variance being based on prior measurements of the same biological variable using the same microarray or two-dimensional gel electrophoresis;
(a) determining, by the computer, a propagated variance based on said predetermined measurement variances {σ t 2 } of said plurality of measurements {y t }, wherein said propagated variance is determined according to the equation
σ
y
_
t
p
2
=
∑
i
=
1
n
t
σ
t
2
n
t
2
;
and
(b) outputting to a user, a user interface device, a computer readable storage medium, or a local or remote computer system; or displaying: a result of said determining step (a) in order to provide the result of the analysis of the plurality of measurements of the biological variable, which is a transcript level of a gene measured using a microarray or an abundance of a protein measured using a protein microarray or two-dimensional gel electrophoresis.