A combina¸c˜ao de valores de prova, usando-os diretamente ou transformando-os (Pestana [3]) ´e simples, mas muitas vezes pouco eficaz, pois quando se tem como objetivo avaliar uma determinada hip´otese nula, ´e muito pouco adequado assumir que ´e v´alida a correspondente hip´otese combinada. Consequentemente, alguns dos valores de prova observados n˜ao devem ser uniformes. Da´ı que os conceitos de valores de prova generalizados e de valores de prova aleat´orios para uma s´ıntese possam ser ´uteis (veja-se Hartung et al. [1] e Kulinskaya et al. [2]).
Xp 1− Xp 6
e consequentemente sempre mais pr´oxima da uniforme X0 (coincidindo com a uniforme se alguma
das vari´aveis originais for uniforme).
Apresentamos tamb´em uma extens˜ao para o caso de dependˆencia auto-regressiva. Usamos aquele resultado para aumentar computacionalmente amostras de valores de prova, discutindo formas de o fazer que n˜ao piorem a potˆencia do teste combinado.
3
Conclus˜ao
Afinal a amplia¸c˜ao computacional das amostras pode n˜ao ser uma boa ideia, pois ´e poss´ıvel que caracter´ısticas estruturais do modelo – em particular, a entropia m´axima da uniforme padr˜ao, na classe das leis com suporte (0,1) – tenham um efeito perverso na potˆencia dos testes que se pretende fazer.
Agradecimentos
Trabalho financiado por fundos nacionais atrav´es da FCT — Funda¸c˜ao para a Ciˆencia e a Tecnologia, no ˆambito do projecto PEst-OE/MAT/UI0006/2011.
Referˆencias
[1] Hartung, J., Knapp, G., Sinha, B.K. (2008). Statistical Meta-Analysis with Applications, Wiley, New York.
[2] Kulinskaya, E., Morgenthaler, S., Staudte, R.G. (2008). Meta Analysis. A Guide to Calibrating and Combining Statistical Evidence, Wiley, Chichester.
[3] Pestana, D. (2011). Combining p-values. In M. Lovric (Ed.), International Encyclopedia of Statistical Science, 1145–1147, Springer Verlag, New York.
I Portuguese Meeting of Biometry & I Portuguese-Galician Meeting of Biometry Oral Communication
Adjusted p-values for SGoF multitesting procedure: Definition and properties Irene Castro Conde
SiDOR Research Group, University of Vigo, Spain, [email protected] Jacobo de U˜na ´Alvarez
SiDOR Research Group; Department of Statistics and OR, University of Vigo, [email protected] Keywords: False discovery rate, Family-wise error rate, Multiple comparisons, Dependent tests. Summary: In the paper Carvajal-Rodr´ıguez, de U˜na- ´Alvarez and Rol´an- ´Alvarez (2009) [1] a new multitest correction named SGoF (from Sequential Goodness-of-Fit) was introduced; this method was extended to possibly correlated tests in de U˜na- ´Alvarez (2012) [4] , who introduced the Beta- Binomial SGoF (BB-SGoF) procedure. Both SGoF and BB-SGoF have the property of increasing their statistical power when increasing the number of tests, which is very useful in omic sciences: genomics, proteomics, etc. because they typically involve the simultaneous testing of hundreds or thousands of hypotheses. Statistical properties and false discovery rate and power levels in practical settings for SGoF-type strategies were further investigated in de U˜na- ´Alvarez (2011, 2012) [3, 4] and Castro-Conde and de U˜na- ´Alvarez (2013) [2]. In this talk we introduce adjusted p-values for SGoF method and we investigate their properties. Time permitting, adjusted p-values for BB-SGoF will be presented and discussed too.
1
Introduction
Nowadays, there exist many statistical inference problems in areas such genomic and protenomics which involve the simultaneous test of hundreds or thousands of null hypotheses producing as a result a number of signicant p-values or effects. Moreover, these hypotheses may have complex and unknown dependence structure among themselves. See e.g. Dudoit and Van der Laan (2008) [5] for an introduction to this area.
One of the main problems in multiple hypotheses testing is that, if one does not take the multiplicity of tests into account, then the probability that some of the true null hypotheses are rejected may be overly large. So, in the multitesting setting, a specific procedure for deciding which null hypotheses should be rejected is needed.
In the paper Carvajal-Rodr´ıguez, de U˜na- ´Alvarez and Rol´an- ´Alvarez [1] a new multitest correction named SGoF (from Sequential Goodness-of-Fit) was introduced; this method was extended to possibly correlated tests in de U˜na- ´Alvarez [4], who introduced the Beta-Binomial SGoF (BB-SGoF) procedure. Both SGoF and BB-SGoF procedures use the p-values to decide which hypotheses are to be rejected. Let us define formally the concept of unadjusted p-value in this multitest setting (Dudoit and Van der Laan [5]).
Definition 1.1 (Unadjusted p-value) The unadjusted p-value pi, for the single test of null hy-
pothesis H0i, is defined as
pi ≡ inf{α ∈ [0, 1] : Reject H0i at single test nominal level α}, i = 1, ..., n.
Definition 1.2 (Adjusted p-value) The adjusted p-value ˜pi, for the test of null hypothesis H0i,
is defined as ˜
pi≡ inf{α ∈ [0, 1] : Reject H0i at multiple comparison procedure nominal level α}
That is, the adjusted p-value ˜pi, for null hypothesis H0i, is the smallest nominal Type I error level
of the multiple hypothesis testing procedure at which one would reject H0i.
Obtaining adjusted p-values consists of determining for each comparison the smallest level of sig- nificance that would result in the comparison being declared significant. As in single hypothesis tests, the smaller the adjusted p-value ˜pi, the stronger the evidence against the corresponding null
hypothesis H0i.
In this talk we introduce adjusted p-values for SGoF method and we investigate their properties. Time permitting, adjusted p-values for BB-SGoF will be presented and discussed too. These method have the particularity that they depends on two parameters, α which is the nominal level of the metatest and controls the FWER weakly and γ which represents the initial significance threshold for significant p-values. In practice, to calculate the adjusted p-values we will considerate that α = γ so we consider there exists only one parameter playing the role of nominal level.
Acknowledgments
Work supported by the Grant MTM2011-23204 (FEDER support included) of the Spanish Ministry of Science and Innovation. Support from the Xunta de Galicia Grant 10PXIB300068PR is also acknowledged.
References
[1] Carvajal-Rodr´ıguez, A., de U˜na- ´Alvarez, J., Rol´an- ´Alvarez, E. (2009). A new multitest correction (SGoF) that increases its statistical power when increasing the number of tests. BMC Bioinformatics 10, 209.
[2] Castro-Conde, I., de U˜na- ´Alvarez, J. (2013). Performance of Beta-Binomial SGoF multitesting method for dependent gene expression levels: a simulation study. Proceedings of Bioinformatics 2013 Interna- tional Conference on Bioinformatics Models, Methods and Algorithms (Pedro Fernandes, Jordi Sole- Casals, Ana Fred and Hugo Gamboa Eds.), SciTePress.
[3] de U˜na- ´Alvarez, J .(2011). On the statistical properties of SGoF multitesting method. Statistical Appli- cations in Genetics and Molecular Biology 10(1), 18.
[4] de U˜na- ´Alvarez, J. (2012). The Beta-Binomial SGoF method for multiple dependent tests. Statistical Applications in Genetics and Molecular Biology 11(3), 14.
[5] Dudoit, S., Van der Laan, M.J. (2008). Multiple Testing Procedures with Applications to Genomics. New York: Springer.
I Encontro Portuguˆes de Biometria & I Encontro Luso-Galaico de Biometria Comunica¸c˜ao Oral
Influˆencia do n´ıvel socioecon´omico da regi˜ao no risco de fratura do f´emur proximal Carla Oliveira
INEB - Instituto de Engenharia Biom´edica, Universidade do Porto; Departamento de Epidemiolo- gia Cl´ınica, Medicina Preditiva e Sa´ude P´ublica, Faculdade de Medicina, Universidade do Porto; ISPUP - Instituto de Sa´ude P´ublica da Universidade do Porto, [email protected]
Denisa Mendon¸ca
ICBAS - Instituto de Ciˆencias Biom´edicas Abel Salazar e ISPUP - Instituto de Sa´ude P´ublica da Universidade do Porto, [email protected]
Maria de F´atima de Pina
INEB - Departamento de Epidemiologia Cl´ınica, Medicina Preditiva e Sa´ude P´ublica, Faculdade de Medicina, ICBAS - Instituto de Ciˆencias Biom´edicas Abel Salazar e ISPUP - Instituto de Sa´ude P´ublica da Universidade do Porto, [email protected]
Palavras–chave: Epidemiologia, Sistemas de Informa¸c˜ao Geogr´afica (GIS), Estat´ıstica, Biometria.
1
Introdu¸c˜ao
As fraturas do f´emur proximal (FFP) s˜ao consideradas um grave problema de sa´ude p´ublica e po- dem estar relacionadas com as condi¸c˜oes sociais e econ´omicas das popula¸c˜oes. Alguns estudos tˆem mostrado que regi˜oes mais desfavorecidas, com maior percentagem de indiv´ıduos com m´a alimen- ta¸c˜ao, atividade f´ısica insuficiente e deficiente acesso ao sistema tendem a ter taxas de incidˆencia de FFP mais elevadas.
2
Objetivo
Avaliar a associa¸c˜ao entre a incidˆencia de FFP e o n´ıvel socioecon´omico (SES) por munic´ıpio, em Portugal.