Multivariable Fractional Polynomials
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Multivariable Fractional Polynomials

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Machine Learning
Published
January 23, 2022
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Multivariable Fractional Polynomials(MFP)๋ž€?

ํฌ๊ท€ ๋ชจ๋ธ์—์„œ ์ค‘์š”ํ•œ ์˜ˆ์ธก ๋ณ€์ˆ˜๋ฅผ ์„ ํƒํ•˜์—ฌ ์˜ˆ์ธก๋ฅ ์„ ๋†’์ด๋Š” ์ „๋žต์„ ์‚ฌ์šฉํ•œ๋‹ค. ๋ชจ๋ธ ๊ตฌ์ถ• ์‹œ ์„ ํ˜• ๋ชจ๋ธ์ด ์‹์„ ์ž˜ ํ‘œํ˜„ํ•  ์ˆ˜ ์žˆ์ง€๋งŒ ์„ ํ˜•์„ฑ ๊ฐ€์ •์ด ์˜ฌ๋ฐ”๋ฅด์ง€ ์•Š์•„ ์ตœ์ข… ๋ชจ๋ธ์ด ์ž˜๋ชป ์ง€์ •๋  ์ˆ˜ ์žˆ๋‹ค. ๊ทธ๋ž˜์„œ ๋น„์„ ํ˜•์„ฑ์ด ํŠน์ง•์ธ MFP๊ฐ€ ์žˆ๋‹ค. MFP๋Š” ๋ถ„์ˆ˜ ๋‹คํ•ญ์‹(fractional polynomials)๊ณผ ํ›„์ง„ ์†Œ๊ฑฐ(backward elimination)์˜ ์กฐํ•ฉ์œผ๋กœ ์ด๋ฃจ์–ด์ ธ ์žˆ๋‹ค. ์—ฌ๊ธฐ์„œ ๋ถ„์ˆ˜ ๋‹คํ•ญ์‹์ด๋ž€ ์ด๋ฆ„ ๊ทธ๋Œ€๋กœ ๋ถ„์ˆ˜๋กœ ์ด๋ฃจ์–ด์ ธ ์žˆ๋Š” ๋‹คํ•ญ์‹์„ ๋œปํ•˜๋ฉฐ, log, ์ •์ˆ˜๊ฐ€ ์•„๋‹Œ ์ง€์ˆ˜(exponentiation), ์ง€์ˆ˜ ๋ฐ˜๋ณต ํ—ˆ์šฉ์˜ 3๊ฐ€์ง€๋ฅผ ํ—ˆ์šฉํ•œ๋‹ค. ์ฆ‰, ์ค‘์š”ํ•œ ๋ณ€์ˆ˜๋ฅผ ์„ ํƒํ•˜๊ณ  ์—ฐ์† ์˜ˆ์ธก ๋ณ€์ˆ˜์— ์ ํ•ฉํ•œ ๊ธฐ๋Šฅ์  ํ˜•ํƒœ๋ฅผ ๊ฒฐ์ •ํ•˜๋Š” ๋‘๊ฐ€์ง€ ๋ชฉํ‘œ๋ฅผ ๊ฐ€์ง„ ๋‹ค๋ณ€์ˆ˜ ๋ชจ๋ธ์„ ๋งŒ๋“œ๋Š” ๊ธฐ๋ฒ•์ด๋‹ค.
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MFP์˜ ์ทจ์•ฝ์ 

MFP๋Š” predictor๋ฅผ ์„ ํƒํ•  ๋•Œ ๋ฐ์ดํ„ฐ์˜ ์ž‘์€ ๋ณ€ํ™”์— ๋งค์šฐ ๋ฏผ๊ฐํ•˜๊ฒŒ ๋ฐ˜์‘ํ•œ๋‹ค. ์ด๋Ÿฐ ๋ถˆ์•ˆ์ •์„ฑ์„ ๊ทน๋ณตํ•˜๊ธฐ ์œ„ํ•ดย bootstap resampling์„ ์ด์šฉํ•œ๋‹ค. ์‚ฌ์šฉ๋ฐฉ๋ฒ•์€ ๊ฐ„๋‹จํ•˜๋‹ค. ๋จผ์ € spearman correlation coefficient๋ฅผ ๊ตฌํ•˜์—ฌ ์˜ˆ์ธก ์ •๋ณด๊ฐ€ ๋ณ€์ˆ˜ ๊ฐ„์— ๊ณต์œ ๋  ๊ฐ€๋Šฅ์„ฑ์„ ํŒŒ์•…ํ•˜๊ณ  ๋†’์€ ์ƒ๊ด€๊ด€๊ณ„๊ฐ€ ์žˆ๋Š” ๋ณ€์ˆ˜๋ฅผ ์ฐพ์•„๋‚ธ๋‹ค. ์ด ๋ณ€์ˆ˜๋Š” ํ–ฅ ํ›„ย ์šฐ์„  ์ˆœ์œ„๋กœ ์ง€์ •๋  ์ˆ˜ ์žˆ๋‹ค.Bootstrap assessment of the stability of multivariable models ๋…ผ๋ฌธ์„ ์ฐธ๊ณ ํ•˜์—ฌ prostate cancer data๋ฅผ ์ด์šฉํ•œ ์‹คํ—˜์„ ์˜ˆ์‹œ๋กœ ๋“ค๊ฒ ๋‹ค. prostrate cancer data๋Š” ์•„๋ž˜์˜ ๋ณ€์ˆ˜๋ฅผ ๊ฐ€์ง€๊ณ  ์žˆ๊ณ  lpsa๋ฅผ ์˜ˆ์ธกํ•˜๋Š” ๊ฒƒ์ด ์ฃผ ํƒœ์Šคํฌ์ด๋‹ค.
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spearman correlation coefficient๋ฅผ ๊ตฌํ–ˆ์„ ๋•Œ lpas์™€ ์ƒ๊ด€๊ด€๊ณ„๊ฐ€ ๋†’์€ ์˜ˆ์ธก ๋ณ€์ˆ˜๋Š” cavol์ด๋‹ค.
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R์˜ mfpboot ํ•จ์ˆ˜๋ฅผ ์‚ฌ์šฉํ•ด์„œ mfp๋ฅผ boostrap์œผ๋กœ 100๊ฐœ๋ฅผ resamplingํ•œ ๊ฒฐ๊ณผ age๋Š” ์ฃผ์š” ๋ณ€์ˆ˜๊ฐ€ ์•„๋‹ˆ๊ณ  svi, cavol(์˜ˆ์ƒํ•œ๋Œ€๋กœ), weight, pgg45๊ฐ€ ์ค‘์š”ํ•œ ๋ณ€์ˆ˜์ด๋‹ค. Obs๋Š” ๋ณ€์ˆ˜๋ฅผ ์ค‘์š”ํ•˜๊ฒŒ ๋ฝ‘์€ ๊ฐฏ์ˆ˜๋ผ๊ณ  ํ•ด์„ํ•˜๋ฉด ๋˜๋Š”๋ฐ cavol์˜ ๊ฒฝ์šฐ 100๊ฐœ ์ค‘์— 100๊ฐœ๋ฅผ ๋ฝ‘์•˜๋‹ค๋Š” ๊ฒƒ์„ ์˜๋ฏธํ•œ๋‹ค.
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