# Sensitivity analysis for sample size given the Accuracy in Parameter Estimation approach for the standardized mean difference.

### Description

Performs sensitivity analysis for sample size determination for the standardized mean difference given a population and a standardized mean difference. Allows one to determine the effect of being wrong when estimating the population standardized mean difference in terms of the width of the obtained (two-sided) confidence intervals.

### Usage

1 2 3 |

### Arguments

`true.delta` |
population standardized mean difference |

`estimated.delta` |
estimated standardized mean difference; can be |

`desired.width` |
describe full width for the confidence interval around the population standardized mean difference |

`selected.n` |
selected sample size to use in order to determine distributional properties of at a given value of sample size |

`assurance` |
parameter to ensure confidence interval width with a specified degree of certainty (must
be |

`certainty` |
an alias for |

`conf.level` |
the desired degree of confidence (i.e., 1-Type I error rate). |

`G` |
number of generations (i.e., replications) of the simulation |

`print.iter` |
to print the current value of the iterations |

`...` |
for modifying parameters of functions this function calls |

### Details

For sensitivity analysis when planning sample size given the desire to obtain narrow confidence intervals
for the population standardized mean difference. Given a population value and an estimated value, one can determine
the effects of incorrectly specifying the population standardized mean difference (`true.delta`

) on the
obtained widths of the confidence intervals. Also, one can evaluate the percent of the confidence intervals
that are less than the desired width (especially when modifying the `certainty`

parameter); see `ss.aipe.smd`

)

Alternatively, one can specify `selected.n`

to determine the results at a particular sample size (when doing this `estimated.delta`

cannot be specified).

### Value

`Results` |
list of the results in |

`Specifications` |
specification of the function |

`Summary` |
summary measures of some important descriptive statistics |

`d` |
contained in |

`Full.Width` |
contained in |

`Width.from.d.Upper` |
contained in |

`Width.from.d.Lower` |
contained in |

`Type.I.Error.Upper` |
contained in |

`Type.I.Error.Lower` |
contained in |

`Type.I.Error` |
contained in |

`Upper.Limit` |
contained in |

`Low.Limit` |
contained in |

`replications` |
contained in |

`true.delta` |
contained in |

`estimated.delta` |
contained in |

`desired.width` |
contained in |

`certainty` |
contained in |

`n.j` |
contained in |

`mean.full.width` |
contained in |

`median.full.width` |
contained in |

`sd.full.width` |
contained in |

`Pct.Less.Desired` |
contained in |

`mean.Width.from.d.Lower` |
contained in |

`mean.Width.from.d.Upper` |
contained in |

`Type.I.Error.Upper` |
contained in |

`Type.I.Error.Lower` |
contained in |

### Note

Returns three lists, where each list has multiple components.

### Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

### References

Cumming, G. & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are
based on central and noncentral distributions, *Educational and Psychological Measurement, 61*, 532–574.

Hedges, L. V. (1981). Distribution theory for Glass's Estimator of effect size and related estimators. *Journal of Educational Statistics, 2*, 107–128.

Kelley, K. (2005). The effects of nonnormal distributions on confidence intervals around the standardized mean
difference: Bootstrap and parametric confidence intervals, *Educational and Psychological Measurement, 65*, 51–69.

Steiger, J. H., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of
statistical methods. In L. L. Harlow, S. A. Mulaik, & J.H. Steiger (Eds.), *What if there were
no significance tests?* (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

### See Also

`ss.aipe.smd`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | ```
# Since 'true.delta' equals 'estimated.delta', this usage
# returns the results of a correctly specified situation.
# Note that 'G' should be large (50 is used to make the example run easily)
# Res.1 <- ss.aipe.smd.sensitivity(true.delta=.5, estimated.delta=.5,
# desired.width=.30, certainty=NULL, conf.level=.95, G=50,
# print.iter=FALSE)
# Lists contained in Res.1.
# names(Res.1)
#Objects contained in the 'Results' lists.
# names(Res.1$Results)
#Extract d from the Results list of Res.1.
# d <- Res.1$Results$d
# hist(d)
# Pull out summary measures
# Res.1$Summary
# True standardized mean difference is .4, but specified at .5.
# Change 'G' to some large number (e.g., G=5,000)
# Res.2 <- ss.aipe.smd.sensitivity(true.delta=.4, estimated.delta=.5,
# desired.width=.30, certainty=NULL, conf.level=.95, G=50,
# print.iter=FALSE)
# The effect of the misspecification on mean confidence intervals is:
# Res.2$Summary$mean.full.width
# True standardized mean difference is .5, but specified at .4.
# Res.3 <- ss.aipe.smd.sensitivity(true.delta=.5, estimated.delta=.4,
# desired.width=.30, certainty=NULL, conf.level=.95, G=50,
# print.iter=FALSE)
# The effect of the misspecification on mean confidence intervals is:
# Res.3$Summary$mean.full.width
``` |

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