sim_log_lognormal: Simulate data from a normal distribution
In depower: Power Analysis for Differential Expression Studies
View source: R/sim_log_lognormal.r
sim_log_lognormal R Documentation
Simulate data from a normal distribution
Description
Simulate data from the log transformed lognormal distribution (i.e. a normal
distribution). This function handles all three cases:
One-sample data
Dependent two-sample data
Independent two-sample data
Usage
sim_log_lognormal(
n1,
n2 = NULL,
ratio,
cv1,
cv2 = NULL,
cor = 0,
nsims = 1L,
return_type = "list",
ncores = 1L,
messages = TRUE
)
Arguments
n1
(integer: [2, Inf))
The sample size(s) of sample 1.
n2
(integer: NULL; [2, Inf))
The sample size(s) of sample 2. Set as NULL if you want to simulate
for the one-sample case.
ratio
(numeric: (0, Inf))
The assumed population fold change(s) of sample 2 with respect to
sample 1.
For one-sample data, ratio is defined as the geometric mean (GM) of the
original lognormal population distribution.
For dependent two-sample data, ratio is defined by
GM(sample 2 / sample 1) of the original lognormal population distributions.
e.g. ratio = 2 assumes that the geometric mean of all paired
ratios (sample 2 / sample 1) is 2.
For independent two-sample data, the ratio is defined by
GM(group 2) / GM(group 1) of the original lognormal population distributions.
e.g. ratio = 2 assumes that the geometric mean of sample 2 is 2
times larger than the geometric mean of sample 1.
See 'Details' for additional information.
cv1
(numeric: (0, Inf))
The coefficient of variation(s) of sample 1 in the original lognormal
data.
cv2
(numeric: NULL; (0, Inf))
The coefficient of variation(s) of sample 2 in the original lognormal
data. Set as NULL if you want to simulate for the one-sample case.
cor
(numeric: 0; [-1, 1])
The correlation(s) between sample 1 and sample 2 in the original
lognormal data. Not used for the one-sample case.
nsims
(Scalar integer: 1L; [1,Inf))
The number of simulated data sets. If nsims > 1, the data is
returned in a list-column of a depower simulation data frame.
return_type
(string: "list"; c("list", "data.frame"))
The data structure of the simulated data. If "list" (default), a
list object is returned. If "data.frame" a data frame in tall format
is returned. The list object provides computational efficiency and the
data frame object is convenient for formulas. See 'Value'.
ncores
(Scalar integer: 1L; [1,Inf))
The number of cores (number of worker processes) to use. Do not set
greater than the value returned by parallel::detectCores(). May be
helpful when the number of parameter combinations is large and nsims
is large.
messages
(Scalar logical: TRUE)
Whether or not to display messages for pathological simulation cases.
Details
Based on assumed characteristics of the original lognormal distribution, data
is simulated from the corresponding log-transformed (normal) distribution.
This simulated data is suitable for assessing power of a hypothesis for the
geometric mean or ratio of geometric means from the original lognormal data.
This method can also be useful for other population distributions which are
positive and where it makes sense to describe the ratio of geometric means.
However, the lognormal distribution is theoretically correct in the sense
that you can log transform to a normal distribution, compute the summary
statistic, then apply the inverse transformation to summarize on the original
lognormal scale.
Let GM(\cdot) be the geometric mean and AM(\cdot) be the
arithmetic mean. For independent lognormal samples X_1 and X_2
\text{Fold Change} = \frac{GM(X_2)}{GM(X_1)}
For dependent lognormal samples X_1 and X_2
\text{Fold Change} = GM\left( \frac{X_2}{X_1} \right)
Unlike ratios and the arithmetic mean, for equal sample sizes of
X_1 and X_2 it follows that
\frac{GM(X_2)}{GM(X_1)} = GM \left( \frac{X_2}{X_1} \right) =
e^{AM(\ln X_2) - AM(\ln X_1)} = e^{AM(\ln X_2 - \ln X_1)}.
The coefficient of variation (CV) for X is defined as
CV = \frac{SD(X)}{AM(X)}
The relationship between sample statistics for the original lognormal data
(X) and the natural logged data (\ln{X}) are
\begin{aligned}
AM(X) &= e^{AM(\ln{X}) + \frac{Var(\ln{X})}{2}} \\
GM(X) &= e^{AM(\ln{X})} \\
Var(X) &= AM(X)^2 \left( e^{Var(\ln{X})} - 1 \right) \\
CV(X) &= \frac{\sqrt{AM(X)^2 \left( e^{Var(\ln{X})} - 1 \right)}}{AM(X)} \\
&= \sqrt{e^{Var(\ln{X})} - 1}
\end{aligned}
and
\begin{aligned}
AM(\ln{X}) &= \ln \left( \frac{AM(X)}{\sqrt{CV(X)^2 + 1}} \right) \\
Var(\ln{X}) &= \ln(CV(X)^2 + 1) \\
Cor(\ln{X_1}, \ln{X_2}) &= \frac{\ln \left( Cor(X_1, X_2)CV(X_1)CV(X_2) + 1 \right)}{SD(\ln{X_1})SD(\ln{X_2})}
\end{aligned}
Based on the properties of correlation and variance,
\begin{aligned}
Var(X_2 - X_1) &= Var(X_1) + Var(X_2) - 2Cov(X_1, X_2) \\
&= Var(X_1) + Var(X_2) - 2Cor(X_1, X_2)SD(X_1)SD(X_2) \\
SD(X_2 - X_1) &= \sqrt{Var(X_2 - X_1)}
\end{aligned}
The standard deviation of the differences gets smaller the more positive
the correlation and conversely gets larger the more negative the correlation.
For the special case where the two samples are uncorrelated and each has the
same variance, it follows that
\begin{aligned}
Var(X_2 - X_1) &= \sigma^2 + \sigma^2 \\
SD(X_2 - X_1) &= \sqrt{2}\sigma
\end{aligned}
Value
If nsims = 1 and the number of unique parameter combinations is
one, the following objects are returned:
If one-sample data with return_type = "list", a list:
Slot Name Description
1 One sample of simulated normal values.
If one-sample data with return_type = "data.frame", a data frame:
Column Name Description
1 item Pair/subject/item indicator.
2 value Simulated normal values.
If two-sample data with return_type = "list", a list:
Slot Name Description
1 Simulated normal values from sample 1.
2 Simulated normal values from sample 2.
If two-sample data with return_type = "data.frame", a data frame:
Column Name Description
1 item Pair/subject/item indicator.
2 condition Time/group/condition indicator.
3 value Simulated normal values.
If nsims > 1 or the number of unique parameter combinations is greater than
one, each object described above is returned in data frame, located in a
list-column named data.
If one-sample data, a data frame:
Column Name Description
1 n1 The sample size.
2 ratio Geometric mean [GM(sample 1)].
3 cv1 Coefficient of variation for sample 1.
4 nsims Number of data simulations.
5 distribution Distribution sampled from.
6 data List-column of simulated data.
If two-sample data, a data frame:
Column Name Description
1 n1 Sample size of sample 1.
2 n2 Sample size of sample 2.
3 ratio Ratio of geometric means [GM(sample 2) / GM(sample 1)]
or geometric mean ratio [GM(sample 2 / sample 1)].
4 cv1 Coefficient of variation for sample 1.
5 cv2 Coefficient of variation for sample 2.
6 cor Correlation between samples.
7 nsims Number of data simulations.
8 distribution Distribution sampled from.
9 data List-column of simulated data.
References
\insertRefjulious_2004depower
\insertRefhauschke_1992depower
\insertRefjohnson_1994depower
See Also
stats::rnorm(), mvnfast::rmvn()
Examples
#----------------------------------------------------------------------------
# sim_log_lognormal() examples
#----------------------------------------------------------------------------
library(depower)
# Independent two-sample data returned in a data frame
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0,
nsims = 1,
return_type = "data.frame"
)
# Independent two-sample data returned in a list
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0,
nsims = 1,
return_type = "list"
)
# Dependent two-sample data returned in a data frame
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0.4,
nsims = 1,
return_type = "data.frame"
)
# Dependent two-sample data returned in a list
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0.4,
nsims = 1,
return_type = "list"
)
# One-sample data returned in a data frame
sim_log_lognormal(
n1 = 10,
ratio = 1.3,
cv1 = 0.35,
nsims = 1,
return_type = "data.frame"
)
# One-sample data returned in a list
sim_log_lognormal(
n1 = 10,
ratio = 1.3,
cv1 = 0.35,
nsims = 1,
return_type = "list"
)
# Independent two-sample data: two simulations for four parameter combinations.
# Returned as a list-column of lists within a data frame
sim_log_lognormal(
n1 = c(10, 20),
n2 = c(10, 20),
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0,
nsims = 2,
return_type = "list"
)
# Dependent two-sample data: two simulations for two parameter combinations.
# Returned as a list-column of lists within a data frame
sim_log_lognormal(
n1 = c(10, 20),
n2 = c(10, 20),
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0.4,
nsims = 2,
return_type = "list"
)
# One-sample data: two simulations for two parameter combinations
# Returned as a list-column of lists within a data frame
sim_log_lognormal(
n1 = c(10, 20),
ratio = 1.3,
cv1 = 0.35,
nsims = 2,
return_type = "list"
)
depower documentation built on April 3, 2025, 9:23 p.m.
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View source: R/sim_log_lognormal.r
| sim_log_lognormal | R Documentation |
Simulate data from a normal distribution
Description
Simulate data from the log transformed lognormal distribution (i.e. a normal distribution). This function handles all three cases:
One-sample data
Dependent two-sample data
Independent two-sample data
Usage
sim_log_lognormal(
n1,
n2 = NULL,
ratio,
cv1,
cv2 = NULL,
cor = 0,
nsims = 1L,
return_type = "list",
ncores = 1L,
messages = TRUE
)
Arguments
n1 |
(integer: |
n2 |
(integer: |
ratio |
(numeric:
See 'Details' for additional information. |
cv1 |
(numeric: |
cv2 |
(numeric: |
cor |
(numeric: |
nsims |
(Scalar integer: |
return_type |
(string: |
ncores |
(Scalar integer: |
messages |
(Scalar logical: |
Details
Based on assumed characteristics of the original lognormal distribution, data is simulated from the corresponding log-transformed (normal) distribution. This simulated data is suitable for assessing power of a hypothesis for the geometric mean or ratio of geometric means from the original lognormal data.
This method can also be useful for other population distributions which are positive and where it makes sense to describe the ratio of geometric means. However, the lognormal distribution is theoretically correct in the sense that you can log transform to a normal distribution, compute the summary statistic, then apply the inverse transformation to summarize on the original lognormal scale.
Let GM(\cdot) be the geometric mean and AM(\cdot) be the
arithmetic mean. For independent lognormal samples X_1 and X_2
\text{Fold Change} = \frac{GM(X_2)}{GM(X_1)}
For dependent lognormal samples X_1 and X_2
\text{Fold Change} = GM\left( \frac{X_2}{X_1} \right)
Unlike ratios and the arithmetic mean, for equal sample sizes of
X_1 and X_2 it follows that
\frac{GM(X_2)}{GM(X_1)} = GM \left( \frac{X_2}{X_1} \right) =
e^{AM(\ln X_2) - AM(\ln X_1)} = e^{AM(\ln X_2 - \ln X_1)}.
The coefficient of variation (CV) for X is defined as
CV = \frac{SD(X)}{AM(X)}
The relationship between sample statistics for the original lognormal data
(X) and the natural logged data (\ln{X}) are
\begin{aligned}
AM(X) &= e^{AM(\ln{X}) + \frac{Var(\ln{X})}{2}} \\
GM(X) &= e^{AM(\ln{X})} \\
Var(X) &= AM(X)^2 \left( e^{Var(\ln{X})} - 1 \right) \\
CV(X) &= \frac{\sqrt{AM(X)^2 \left( e^{Var(\ln{X})} - 1 \right)}}{AM(X)} \\
&= \sqrt{e^{Var(\ln{X})} - 1}
\end{aligned}
and
\begin{aligned}
AM(\ln{X}) &= \ln \left( \frac{AM(X)}{\sqrt{CV(X)^2 + 1}} \right) \\
Var(\ln{X}) &= \ln(CV(X)^2 + 1) \\
Cor(\ln{X_1}, \ln{X_2}) &= \frac{\ln \left( Cor(X_1, X_2)CV(X_1)CV(X_2) + 1 \right)}{SD(\ln{X_1})SD(\ln{X_2})}
\end{aligned}
Based on the properties of correlation and variance,
\begin{aligned}
Var(X_2 - X_1) &= Var(X_1) + Var(X_2) - 2Cov(X_1, X_2) \\
&= Var(X_1) + Var(X_2) - 2Cor(X_1, X_2)SD(X_1)SD(X_2) \\
SD(X_2 - X_1) &= \sqrt{Var(X_2 - X_1)}
\end{aligned}
The standard deviation of the differences gets smaller the more positive the correlation and conversely gets larger the more negative the correlation. For the special case where the two samples are uncorrelated and each has the same variance, it follows that
\begin{aligned}
Var(X_2 - X_1) &= \sigma^2 + \sigma^2 \\
SD(X_2 - X_1) &= \sqrt{2}\sigma
\end{aligned}
Value
If nsims = 1 and the number of unique parameter combinations is
one, the following objects are returned:
If one-sample data with
return_type = "list", a list:Slot Name Description 1 One sample of simulated normal values. If one-sample data with
return_type = "data.frame", a data frame:Column Name Description 1 itemPair/subject/item indicator. 2 valueSimulated normal values. If two-sample data with
return_type = "list", a list:Slot Name Description 1 Simulated normal values from sample 1. 2 Simulated normal values from sample 2. If two-sample data with
return_type = "data.frame", a data frame:Column Name Description 1 itemPair/subject/item indicator. 2 conditionTime/group/condition indicator. 3 valueSimulated normal values.
If nsims > 1 or the number of unique parameter combinations is greater than
one, each object described above is returned in data frame, located in a
list-column named data.
If one-sample data, a data frame:
Column Name Description 1 n1The sample size. 2 ratioGeometric mean [GM(sample 1)]. 3 cv1Coefficient of variation for sample 1. 4 nsimsNumber of data simulations. 5 distributionDistribution sampled from. 6 dataList-column of simulated data. If two-sample data, a data frame:
Column Name Description 1 n1Sample size of sample 1. 2 n2Sample size of sample 2. 3 ratioRatio of geometric means [GM(sample 2) / GM(sample 1)] or geometric mean ratio [GM(sample 2 / sample 1)]. 4 cv1Coefficient of variation for sample 1. 5 cv2Coefficient of variation for sample 2. 6 corCorrelation between samples. 7 nsimsNumber of data simulations. 8 distributionDistribution sampled from. 9 dataList-column of simulated data.
References
\insertRefjulious_2004depower
\insertRefhauschke_1992depower
\insertRefjohnson_1994depower
See Also
stats::rnorm(), mvnfast::rmvn()
Examples
#----------------------------------------------------------------------------
# sim_log_lognormal() examples
#----------------------------------------------------------------------------
library(depower)
# Independent two-sample data returned in a data frame
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0,
nsims = 1,
return_type = "data.frame"
)
# Independent two-sample data returned in a list
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0,
nsims = 1,
return_type = "list"
)
# Dependent two-sample data returned in a data frame
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0.4,
nsims = 1,
return_type = "data.frame"
)
# Dependent two-sample data returned in a list
sim_log_lognormal(
n1 = 10,
n2 = 10,
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0.4,
nsims = 1,
return_type = "list"
)
# One-sample data returned in a data frame
sim_log_lognormal(
n1 = 10,
ratio = 1.3,
cv1 = 0.35,
nsims = 1,
return_type = "data.frame"
)
# One-sample data returned in a list
sim_log_lognormal(
n1 = 10,
ratio = 1.3,
cv1 = 0.35,
nsims = 1,
return_type = "list"
)
# Independent two-sample data: two simulations for four parameter combinations.
# Returned as a list-column of lists within a data frame
sim_log_lognormal(
n1 = c(10, 20),
n2 = c(10, 20),
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0,
nsims = 2,
return_type = "list"
)
# Dependent two-sample data: two simulations for two parameter combinations.
# Returned as a list-column of lists within a data frame
sim_log_lognormal(
n1 = c(10, 20),
n2 = c(10, 20),
ratio = 1.3,
cv1 = 0.35,
cv2 = 0.35,
cor = 0.4,
nsims = 2,
return_type = "list"
)
# One-sample data: two simulations for two parameter combinations
# Returned as a list-column of lists within a data frame
sim_log_lognormal(
n1 = c(10, 20),
ratio = 1.3,
cv1 = 0.35,
nsims = 2,
return_type = "list"
)
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