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Generating Cohen's Effect Size "h"
Via Arcsin / Arcsine Transformations
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Adolescent Gay / Lesbian Suicide Risk

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Suicidality Studies Index: All Studies (This Page): The Index. - All Random & Special Sample Studies. - All American & Canadian Studies. - All European Studies. - Transgender Studies. - The Results of Additional School-Based North American Youth Risk Behavior Surveys or Similar Surveys - Random Sampling - are Located at Another Location.

Other Pages: Homosexually Oriented People Are Generally at Greater Risk for the More Serious Suicidal Behaviors. - "Attempting Suicide" as Related To Gender Nonconformity & Transgender Issues. - Bell & Weinberg (1978) Homosexualities Study: "Attempted Suicide" Study Results.

Special Section: The 2013 Paper, "Suicide Risk and Sexual Orientation: A Critical Review," Reverses the Conclusions of Two Previously Published Papers. The Re-Analysis - Including Many Meta-Analyses & Using Unconditional Tests for Statistical Significance - Indicates that "Gay/Lesbian/Bisexual Adolescents Are at Risk for Suicide." - In Addition, Expanding the "At Risk" Category to Include Adolescents Known to Only Have Been Harassed/Abused - Because They Were Assumed to be Gay/Lesbian - Produces More Conclusive Results, Especially Applying for Males. This Category Represents "An Expanded Homosexuality Factor in Adolescent Suicide." - Associated Pages: Constructing "The Gay Youth Suicide Myth": Thirty Years of Resisting a Likely Truth & Generating Cohen's Effect Size "h" Via Arcsin / Arcsine Transformations (This Page).

Cohen's Effect Size "h": The Arcsin / Arcsine Transformation of Probabilities

Effect Sizes From the Arcsin Transformation of the Probabilities - Excerpts From Jacob Cohen (1988) 1
Formula Calculations

Φ1 = Φ50% = 2(arcsin √ 0.50) = 2(arcsin 0.7071) = 2(0.7854) = 1.571

Φ2 = Φ25% = 2(arcsin √ 0.25) = 2(arcsin 0.5000) = 2(0.5236) = 1.047

Cohen's Effect Size [ES] h = Φ1 - Φ2 = 1.571 - 1.407 = 0.524

The arcsin for 0.7071 is the sin-1 for 0.7071 in Radians = 0.7854

1. Cohen, Jacob (2008). Statistical power analysis for the behavioral sciences. Second Edition. Hillsdale, New Jersey: Lawrence Erlbaum Associatesm Inc.. Google Books. Amazon.

Table 6.2.1 From Jacob Cohen (1988) 1
1. Cohen, Jacob (2008). Statistical power analysis for the behavioral sciences. Second Edition. Hillsdale, New Jersey: Lawrence Erlbaum Associates, Inc.. Google Books. Amazon.

Table 6.2.2 From Jacob Cohen (1988) 1
Φ = arcsin √ P
1. Cohen, Jacob (2008). Statistical power analysis for the behavioral sciences. Second Edition. Hillsdale, New Jersey: Lawrence Erlbaum Associatesm Inc.. Google Books. Amazon.

Meta-Analysis Produced Results, Formula Calculated Arcsine Differences & Cohen's Related Arcsin
Effect Sizes "h": Two Studies
1, 2
Counts, Percentages, Proportions
Shaffer et al. (1995) 1
[ 3 / 120 = 2.5% = 0.025 = P1] vs. [0 / 147 = 0% = 0.0 = P2]
Renaud et al. (2010) 2 [ 4 / 55 = 7.27% = 0.727 = P1] vs. [0 / 55 = 0% = 0.0 = P2]
For the Arcsin Difference Meta-Analyses using Arcsin Transformations, the Arcsin Difference for each study - specified as "AS" - is calculated using the formula:
AS = arcsin P1 - arcsin P2
AS Results generated via meta-analysis by Schwarzer G (2012) Program in R
AS: Arcsin Difference
Cohen's Effect Size - h, 95% CI
Shaffer et al. (1995) 0.16 (0.04, 0.28)
One-/Two-Tailed p = 0.032, 0.057
0.32 (0.08, 0.56)
Renaud et al. (2010) 0.27 (0.09, 0.46)
One-/Two-Tailed p = 0.021, 0.043 3
0.54 (0.18, 0.92)
Two Studies Combined 0.19 (0.09, 0.30) 0.38 (0.18, 0.60)


AS = arcsin √P1 - arcsin √P2 h = 2(arcsin √P1) - 2(arcsin √P2)
Shaffer et al. (1995) arcsin √0.025 - arcsin √0.00
arcsin 0.50 - arcsin 0.00
0.159 - 0.0 = 0.16
2(arcsin √0.025) - 2(arcsin √0.00)
2(arcsin 0.50) - 2( arcsin 0.00)
2(0.159) - 2(0.0) = 0.318 = 0.32
Renaud et al. (2010) arcsin √0.0727 - arcsin √0.00
arcsin 0.2696 - arcsin 0.00
0.2730 - 0.0 = 0.27
2(arcsin √0.0727) - 2(arcsin √0.00)
2(arcsin 0.2696) - 2( arcsin 0.00)
2(0.273) - 2(0.0) = 0.546 = 0.55

1. Shaffer, D., Fisher, P., Parides, M., Gould, M (1995)
. Sexual orientation in adolescents who commit suicide. Suicide and Life-Threatening Behavior, Supplement 25: 64-71.
PubMed Abstract.
2. Renaud J, Berlim MT, Begolli M, McGirr A, Turecki G (2010). Sexual orientation and gender identity in youth suicide victims: an exploratory study. Canadian Jourrnal of Psychiatry, 55(1): 29-34. PubMed Abstract. Full Text.
3. Calculated with SMP Program, Version 2.1

Using Small Letter "Phi" in Effect Size Equation

Hojat & Xu (2004): Using the small letter phi in Effect Size equation.
φ45% = 2(arcsin √ 0.45) = 2(arcsin 0.67) = 2(0.74) = 1.48
φ37% = 2(arcsin √ 0.37) = 2(arcsin 0.61) = 2(0.65) = 1.30
ES = φ45% − φ37% = 1.48 − 1.30 = 0.18
Here, arcsin √P is the inverse trigonometric function of sin√p[sin-1√P].

φ - small letter phi  ---  Φ - capital letter phi

From pp. 246, 248: Hojat M, Xu G (2004). A Visitor's Guide to Effect Sizes – Statistical Significance Versus Practical (Clinical) Importance of Research Findings. Advances in Health Sciences Education, 9(3): 241-249. Abstract.

Effect Size Magnitudes

Cohen's d / h & Odds Ratios: Effect Size Magnitudes
Effect Size Magnitude
Cohen's d Equivalent
Cohen's h Equivalent 0.20 0.50 0.80
When 0.0% in Control Group
Using Arcsine Transformation
ORs Not Possible
**Shaffer et al.
Cohen's h = 0.32
**Renaud et al.
Cohen's h = 0.55

When 0.5% in Control Group ?
3.5 - 4.0 ?
7 - 8 ?
*ORs: When 1% in Control Group 1.68
Given for 2% to 4%

*ORs: When 5% in Control Group 1.52
2.74 4.72
Given for 6% to 9%

*ORs: When 10% in Control Group 1.46
Given as Minimums by Ferguson (2009): 'Interpret with caution.'
* Given by Chen et al. (2010) & "Cohen's h" Additions by Webpage Authors
** Cohen's h is Effect Size Measure for
Arcsine Transformed Proportion Difference

Arcsine Transformation & Effect Size Bibliography

Our calculations indicate that OR = 1.68, 3.47, and 6.71 are equivalent to Cohen’s d = 0.2 (small), 0.5 (medium), and 0.8 (large), respectively, when disease rate is 1% in the nonexposed group; Cohen’s d < 0.2 when OR < 1.5, and Cohen’s d > 0.8 when OR > 5. It would be useful to values with corresponding qualitative descriptors that estimate the strength of such associations; however, to date there is no consensus as to what those values of OR may be. Cohen (1988) suggested that d = 0.2, 0.5, and 0.8 are small, medium, and large on the basis of his experience as a statistician, but he also warned that these were only “rules of thumb.” Better guidelines are needed to draw conclusions about strength of associations in studies of risks for disease when we use OR as the index of effect size in epidemiological studies. (p. 864)
Abstract: For clinical trials with binary endpoints there are a variety of effect measures, for example risk difference, risk ratio and odds ratio (OR). The choice of metric is not always straightforward and should reflect the clinical question. Additional issues arise if the event of interest is rare. In systematic reviews, trials with zero events in both arms are encountered and often excluded from the meta-analysis.The arcsine difference (AS) is a measure which is rarely considered in the medical literature. It appears to have considerable promise, because it handles zeros naturally, and its asymptotic variance does not depend on the event probability.This paper investigates the pros and cons of using the AS as a measure of intervention effect. We give a pictorial representation of its meaning and explore its properties in relation to other measures. Based on analytical calculation of the variance of the arcsine transformation, a more conservative variance estimate for the rare event setting is proposed. Motivated by a published meta-analysis in cardiac surgery, we examine the statistical properties of the various metrics in the rare event setting.We find the variance estimate of the AS to be more stable than that of the log-OR, even if events are rare. However, parameter estimation is biased if the groups are markedly unbalanced. Though, from a theoretical viewpoint, the AS is a natural choice, its practical use is likely to continue to be limited by its less direct interpretation.

"The arcsine transformation was introduced in the statistical literature for its approximative variance-stabilizing property. The key advantage is that a stabilized variance also leads to more robust estimation. If the risks in the treatment arms are estimated with noise, the variance estimate of the AS is less dramatically changed than that of the log-OR, even if events are rare. This is an advantage of the AS as a measure of treatment effect particularly when zero cell studies occur. A disadvantage is that if events are rare in both groups and the groups sizes are markedly unbalanced, bias will be induced by the transformation. In this situation, though, other methods are likewise prone to bias. (p. 735)
Abstract: In meta-analyses, it sometimes happens that smaller trials show different, often larger, treatment effects. One possible reason for such 'small study effects' is publication bias. This is said to occur when the chance of a smaller study being published is increased if it shows a stronger effect. Assuming no other small study effects, under the null hypothesis of no publication bias, there should be no association between effect size and effect precision (e.g. inverse standard error) among the trials in a meta-analysis.A number of tests for small study effects/publication bias have been developed. These use either a non-parametric test or a regression test for association between effect size and precision. However, when the outcome is binary, the effect is summarized by the log-risk ratio or log-odds ratio (log OR). Unfortunately, these measures are not independent of their estimated standard error. Consequently, established tests reject the null hypothesis too frequently. We propose new tests based on the arcsine transformation, which stabilizes the variance of binomial random variables. We report results of a simulation study under the Copas model (on the log OR scale) for publication bias, which evaluates tests so far proposed in the literature. This shows that: (i) the size of one of the new tests is comparable to those of the best existing tests, including those recently published; and (ii) among such tests it has slightly greater power, especially when the effect size is small and heterogeneity is present. Arcsine tests have additional advantages that they can include trials with zero events in both arms and that they can be very easily performed using the existing software for regression tests.
The arcsine difference has a long history, dating back to the 1940s [55, 59, 117, 118, 119], and is often used in other contexts [58, 120, 121], but not, to our knowledge, as a measure of treatment effect in clinical trials. It is nevertheless briefly mentioned in this context in a series of references [16, 23, 56, 122, 123, 124]. Its attraction is that the arcsine transformation is the asymptotically variance-stabilising transformation for the binomial distribution." (p. 78) "Transforming the binomial risk introduces bias, which will be greater for small sample sizes and rare events." (p.86)

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