Logistic Regression

Biostatistics Logistic regression

Burkhardt Seifert & Alois Tschopp

Biostatistics Unit University of Zurich

Master of Science in Medical Biology 1 Logistic regression

Great importance for medical research

So far: “ordinary” regression

explain an “outcome” variable y through explanatory variables x1,..., xk quantitative outcome variable y (normally distributed) relation usually assumed to be linear

New with logistic regression: outcome y is binary

Master of Science in Medical Biology 2 Examples

A. y = patients survive (y = 0) or die (y =1) x1 = therapy (x1 = A, B; nominal) x2 = age (in years; continuous) x3, . . . = laboratory parameters.

B. case–control–study (epidemiology) y = case (y = 1) or control (y = 0) x1 = exposed (x1 = 1) or not (x1 = 0) x2, . . . = confounder.

Statistical analysis with one independent variable x also: Mann–Whitney test (or unpaired t–test) Fisher‘s exact test (or χ2–test)

Master of Science in Medical Biology 3 Int. J. Cancer: 121, 1764–1770 (2007) ' 2007 Wiley-Liss, Inc.

Consistent expression of the stem cell renewal factor BMI-1 in primary and metastatic melanoma Daniela Mihic-Probst1*, Ariana Kuster1, Sandra Kilgus1, Beata Bode-Lesniewska1, Barbara Ingold-Heppner1, Carly Leung1, Martina Storz1, Burkhardt Seifert2, Silvia Marino3, Peter Schraml1, Reinhard Dummer4 and Holger Moch1 1Department of Pathology, Institute of Surgical Pathology, University Hospital Zurich, Zurich, Switzerland 2Department of Biostatistics, University of Zurich, Zurich, Switzerland 3 Int. J. Cancer: 121, 1764–1770 (2007) Institute of Pathology, Barts and the London, Queen Mary School of Medicine and Dentistry, London, United Kingdom ' 4Department of Dermatology, University Hospital Zurich, Zurich, Switzerland ' 2007 Wiley-Liss, Inc.

Consistent expression of the stem cell renewalStem cell-like factor cells BMI-1 have recently been identified in melanoma cell nancies.8,9 p14Arf promotes cell cycle arrest and apoptosis via its Consistent expression of the stem cell renewallines, but factor their relevance BMI-1 for melanoma pathogenesis is controver- regulation of p53 stability stages. p14Arf effectively prevents the in primary and metastatic melanoma melanoma sial. To characterize the stem cell signature of melanoma, expres- degradation of the tumor suppressor protein p53, which is required sion of stem cell markers BMI-1 and nestin was studied in 64 cuta- for cell cycle arrest.10,11 1* 1 1 neous melanomas,1 165 melanoma metastases as1 well as 53 mela-1 Daniela Mihic-Probst1*,, Ariana Ariana Kuster Kuster1,, Sandra Sandra Kilgus Kilgus1,, Beata Beata Bode-Lesniewska Bode-Lesniewska1,, Barbara Barbara Ingold-Heppner Ingold-Heppner1,, Carly Carly Leung Leung,1, In addition to their tumor suppressor activities, p16ink4a and 1 2 3 noma1 cell lines. Stem4 cell renewal factor1 BMI-1 is a Arf 12,13 Martina Storz1, Burkhardt Burkhardt Seifert Seifert2,, Silvia Silvia Marino Marino3,, Peter Peter Schraml Schramltranscriptional1,, Reinhard Reinhard Dummer repressor Dummer4 ofandand the Holger Holger Ink4a/Arf Moch Moch locus1 encoding p16ink4a 14 are modulators of senescence. Senescence is a perma- 1 Arf 1Department of Pathology, Institute of Surgical Pathology, Universityand Hospital p14 . Increased Zurich, Zurich, nuclear Switzerland BMI-1 expression was detectable in nent growth arrest triggered by the accumulation of cell division, 2Department of Pathology, Institute of Surgical Pathology, University Hospital Zurich, Zurich, Switzerland ink4a Arf 2Department of Biostatistics, University of Zurich, Zurich, Switzerland41 of 64 (64%) primary melanomas, 117 of 165 melanoma metas- leading to telomeric loss. By impacting on p16 and p14 3Department of Biostatistics, University of Zurich, Zurich, Switzerland 3Institute of Pathology, Barts and the London, Queen Mary Schooltases of Medicine (71%) and and 15 Dentistry, of 53 (28%) London, melanoma United cell Kingdom lines. High nestin simultaneously, BMI-1 plays a pivotal role in the regulation of the 4Institute of Pathology, Barts and the London, Queen Mary Schoolexpression of Medicine was and observed Dentistry, in 14 London, of 56 primary United Kingdom melanomas (25%), 4Department of Dermatology, University Hospital Zurich, Zurich, Switzerland cell cycle progression, arrest and senescence. Since BMI-1 regu- Department of Dermatology, University Hospital Zurich, Zurich, Switzerland84 of 165 melanoma metastases (50%) and 21 of 53 melanoma cell lates the Ink4a/Arf locus, which is often deleted or silenced in lines (40%). There was a significant correlation between BMI-1 p 5 p 5 tumors, it is not surprising that BMI-1 overexpression has been and nestin8,9 expressionArf in cell lines ( 0.001) and metastases ( 14 Stem cell-like cells have recently been identified identified in in melanoma melanoma cell cell 0.02).nancies. These8,9 p14 dataArf indicatepromotes that cell cells cycle in arrest primary and apoptosismelanomasvia andits found in various cancer, such as colorectal cancer, nonsmall cell lines, but their relevance for melanoma pathogenesis is controver- nancies. p14 promotes cell cycleArf arrest and apoptosis via its 15 16 17 lines, but their relevance for melanoma pathogenesis is controver- theirregulation metastases of p53 may stability have stem stages. cell p14 properties.Arfeffectively Cell lines prevents obtained the lung cancer, hepatocellular carcinoma, prostate cancer and sial. To characterize the stem cell signature of melanoma, expres- regulation of p53 stability stages. p14 effectively prevents the 18 sial. To characterize the stem cell signature of melanoma, expres- fromdegradation melanoma of the metastases tumor suppressor showed protein a significant p53, which higher is required BMI-1 neuroblastoma. sion of stem cell markers BMI-1 and nestin was studied in 64 cuta- degradation of the tumor10,11 suppressor protein p53, which is required expressionfor cell cycle compared arrest. 10,11 to cell lines from primary melanoma (p 5 neoussion of melanomas, stem cell markers 165 melanoma BMI-1 and metastases nestin was as studied well as in 53 64 mela- cuta- for cell cycle arrest. BMI-1 has been recently shown to play a crucial role in self- ink4a 19,20 neous melanomas, 165 melanoma metastases as well as 53 mela- 0.001).In addition Further, to primary their tumor melanoma suppressor lacking activities, lymphatic p16 metastasesink4a and renewal of stem cells. Stem cells are slowly proliferating cells noma cell lines. Stem cell renewal factor BMI-1 is a ArfIn addition to theirn 5 tumor suppressor12,13 activities, p16 and noma cell lines. Stem cell renewal factor BMI-1 isink4a a at14 presentationArf are modulators (pN0, of senescence.40) was less12,13 frequentlySenescence BMI-1 is a positive perma- transcriptional repressor of the Ink4a/Arf locus encoding p16 ink4a 14 are modulators of senescence. Senescence is a perma-n 5 that give rise to more rapidly proliferating cells that eventually andtranscriptional p14Arf. Increased repressor nuclear of the BMI-1 Ink4a/Arf expression locus wasencoding detectable p16 in thannent melanomas growth arrest presenting triggered with by the lymphatic accumulation metastases of cell (pN1; division, Arf 24;nent 52% growthversus arrest83%; triggeredp 5 0.01). by the Therefore, accumulation BMI-1 of cell expression division, fully differentiate into mature tissue components. Cells with stem 41and of p14 64 (64%). Increased primary nuclear melanomas, BMI-1 117 expression of 165 melanoma was detectable metas- in leading to telomeric loss. By impacting on p16ink4a and p14Arf 41 of 64 (64%) primary melanomas, 117 of 165 melanoma metas- appearsleading to to induce telomeric a metastatic loss. By tendency. impacting Because on p16 BMI-1ink4a and functions p14Arf cell-like features have been identified in a number of human tases (71%) and 15 of 53 (28%) melanoma cell lines. High nestin simultaneously, BMI-1 plays a pivotal role in the regulationink4a of the 21-25 26,27 tases (71%) and 15 of 53 (28%) melanoma cell lines. High nestin assimultaneously, a transcriptional BMI-1 repressor plays of a pivotal the Ink4a/Arf role in the locus, regulation p16 ofand the malignancies including melanoma. . Expression of the expression was observed in 14 of 56 primary melanomas (25%), cellArf cycle progression, arrest and senescence. Since BMI-1 regu-ink4a 84expression of 165 melanoma was observed metastases in 14 (50%)of 56 primary and 21 of melanomas 53 melanoma (25%), cell p14cell cycleexpression progression, was also arrest analyzed. and senescence. A high BMI-1/low Since BMI-1 p16 regu- neural stem cell marker nestin has been demonstrated in mela- 84 of 165 melanoma metastases (50%) and 21 of 53 melanoma cell expressionlates the Ink4a/Arf pattern was locus, a significant which is often predictor deleted of ormetastasis silenced by in noma28 and melanoma cell lines.26 Nestin is an intermediate fila- lines (40%). There was a significant correlation between BMI-1 lates the Ink4a/Arf locus, which is oftenp 5 deleted or silenced in andlines nestin (40%). expression There was in cell a significant lines (p 5 correlation0.001) and metastases between BMI-1 (p 5 meanstumors, of it logistic is not surprisingregression that analysis BMI-1 ( overexpression0.005). This has suggests been ment expressed in neural crestic cells and proliferating neuronal p 5 p 5 tumors, it is not surprising that BMI-1ink4a overexpression14 has been 29,30 and nestin expression in cell lines ( 0.001) and metastases ( thatfound BMI-1 in various mediated cancer, repression such as colorectal of p16 cancer,may contribute14nonsmall to cell an 0.02). These data indicate that cells in primary melanomas and found in various15 cancer, such as colorectal16 cancer, nonsmall17 cell progenitor cells, but not in the adult brain. their0.02). metastases These data may indicate have stem that cell cells properties. in primary Cell melanomas lines obtained and increasedlung cancer, aggressive15 hepatocellular behavior of carcinoma, stem cell-like16prostate melanoma cancer cells.17and their metastases may have stem cell properties. Cell lines obtained 'lung2007 cancer, Wiley-Liss,18hepatocellular Inc. carcinoma, prostate cancer and Overexpression of BMI-1 in stem cells could lead to uncon- from melanoma metastases showed a significant higher BMI-1 neuroblastoma. 18 31 from melanoma metastases showed a significant higher BMI-1 neuroblastoma. trolled growth and cancer. Absence of BMI-1 in hematopoietic expression compared to cell lines from primary melanoma (p 5 BMI-1 has been recently shown to play a crucial role in self- expression compared to cell lines from primary melanoma (p 5 ink4a Arf stem cells leads to an accelerated differentiation, while overex- 0.001). Further, primary melanoma lacking lymphatic metastases KeyBMI-1 words: hasmelanoma; been recently19,20 BMI-1; shown nestin; to p16 play a; p14 crucial; dermal role in nevi; self- 31-33 0.001). Further, primaryn 5 melanoma lacking lymphatic metastases renewal of stem cells. 19,20Stem cells are slowly proliferating cells pression promotes its self-renewal. The impaired hematopoi- at presentation (pN0, n 5 40) was less frequently BMI-1 positive skin;thatrenewal stem give of cells; rise stem to immunohistochemistry cells. more rapidlyStem proliferating cells are slowly cells proliferating that eventually cells thanat presentation melanomas (pN0, presenting40) with was lymphatic less frequently metastases BMI-1 (pN1; positiven 5 that give rise to more rapidly proliferating cells that eventually etic stem cell self-renewal capacity is caused in part by the dere- than melanomasversus presentingp 5 with lymphatic metastases (pN1; n 5 fully differentiate into mature tissue components. Cells with stem ink4a Arf 24; 52% versus 83%; p 5 0.01). Therefore, BMI-1 expression fully differentiate into mature tissue components. Cells with stem pression of p16 and p14 senescence pathways. Thus, BMI- appears24; 52% to induce83%; a metastatic0.01). tendency. Therefore, Because BMI-1 BMI-1 expression functions cell-like features have been identified in a number of human 1 is, at least in part, required for the maintenance of adult stem appears to induce a metastatic tendency. Because BMI-1 functionsink4a cell-likeB-cell-specific features21-25Moloney have been murine identified leukaemia26,27 in virus a number Integration of human site as a transcriptional repressor of the Ink4a/Arf locus, p16 ink4a and malignancies 21-25 including melanoma. 26,27. Expression of the cells in some tissues because of its function in repressing senes- as aArf transcriptional repressor of the Ink4a/Arf locus, p16 ink4aand malignancies including melanoma. . Expression of the Masterp14 ofArf Scienceexpression in Medical was also Biology analyzed. A high BMI-1/low p16 ink4a 1neural(BMI-1) stem was cell first marker identified nestin by retroviral has been insertional demonstrated mutagenesis in mela- cence4 and cell death. Recently, we have shown the relevance of p14 expression was also analyzed. A high BMI-1/low p16 neural28 stem cell marker nestin26 has been demonstratedl in mela- expression pattern was a significant predictor of metastasis by whennoma assessing28 and melanoma collaborating cell lines. oncogenes26Nestin inis an E intermediate-myc transgenic fila- expression pattern was a significant predictorp 5 of metastasis by noma1,2 and melanoma cell lines. Nestin is an intermediate fila- BMI-1 expression for medulloblastoma, a tumor derived from pro- means of logistic regression analysis (p 5 0.005). This suggests mice.ment expressedThe mouse in neuralBMI-1 crestic gene encodes cells and a nuclear proliferating protein neuronal of 45– 19 thatmeans BMI-1 of logistic mediated regression repression analysis of p16 (ink4a may0.005). contribute This suggests to an ment expressed in neural crestic cells3 and29,30 proliferating neuronal genitor cells of the external granular layer of the cerebellum. that BMI-1 mediated repression of p16ink4a may contribute to an 47progenitor kDa consisting cells, but of not 324 in amino the adult acids. brain.Its29,30 3.6 kb cDNA is com- Since melanomas and benign nevi are derived from neuroectoder- increased aggressive behavior of stem cell-like melanoma cells. posedprogenitor of 10 cells, exons but extended not in the over adult 10 brain. kb of genomic DNA. The 'increased2007 Wiley-Liss, aggressive Inc. behavior of stem cell-like melanoma cells. Overexpression of BMI-1 in stem cells could lead to uncon- mal cells in the skin, we aimed to assess whether BMI-1 is also ' humanOverexpression BMI-1 protein of BMI-1structurally31 in stem andcells with couldrespect lead to its to amino uncon- 2007 Wiley-Liss, Inc. trolled growth and cancer. 31Absence of BMI-1 in hematopoietic expressed by melanoma cells. acidtrolled composition growth and is identical cancer. toAbsence the mouse of BMI-1 proteinin hematopoietic and was ink4a Arf stem cells leads to an accelerated differentiation, while overex- Consequently, we analyzed benign nevi, primary melanomas, Key words: melanoma; BMI-1; nestin; p16 ink4a; p14 Arf; dermal nevi; thestem first cells member leads of to the an Polycomb accelerated gene31-33 differentiation, family identified while in overex- mam- Key words: melanoma; BMI-1; nestin; p16 ; p14 ; dermal nevi; pression promotes its self-renewal. 31-33The impaired hematopoi- skin; stem cells; immunohistochemistry mals.pression4 Low promotes BMI-1 itsexpression self-renewal. is observedThe in impaired almost all hematopoi- tissues, melanoma metastases and cell lines, and demonstrate frequent skin; stem cells; immunohistochemistry etic stem cell self-renewal capacity is caused in part by the dere- expression of the stem cell markers BMI-1 and nestin. Moreover, withpressionetic higher stem of cell expressionp16 self-renewalink4a and in p14 the capacityArf brain,senescence spinal is caused cord, pathways. in kidney, part Thus, by lung the BMI- dere- and 3,5 ink4a Arf we show that high BMI-1 expression is associated with an gonads.1pression is, at least ofIn p16 addition,in part,and required BMI-1 p14 forissenescence highly the maintenance expressed pathways. inof Thus,embryonic adult stemBMI- B-cell-specific Moloney murine leukaemia virus Integration site stem1 is, cells at least and thein part, placenta. required1 for the maintenance of adult stem increased risk of metastatic disease. Besides being of prognostic B-cell-specific Moloney murine leukaemia virus Integration site cells in some tissues because of its function in repressing senes- relevance, these data may provide new insights in melanoma tu- 1 (BMI-1) was first identified by retroviral insertional mutagenesis cencecellsBMI-1 in and some is cell atranscriptional tissues death. Recently,because repressor of we its have function of shown the in Ink4a/Arf therepressing relevance locus, senes- of6 when1 (BMI-1) assessing was first collaborating identified by oncogenes retroviral insertionalin El-myc mutagenesis transgenic cence and cell death. Recently, we have shown the relevance of morigenesis. 1,2 l whichBMI-1 encodesexpression 2 for separate medulloblastoma, tumor suppressor a tumor derived genes, from namely pro- when assessing collaborating oncogenes in E -myc transgenic 19 mice.1,2 The mouse BMI-1 gene encodes a nuclear protein of 45– BMI-1ink4a expressionArf for medulloblastoma,Arf a tumor derived from pro- 3 p16genitorand cells p14 of the(p19 externalingranular mouse), which layer of are the read cerebellum. in alternat-19 47mice. kDa consistingThe mouse of BMI-1 324 amino gene acids.encodesIts a nuclear3.6 kb cDNA protein is of com- 45– ink4a 3 ingSincegenitor reading melanomas cells frames. of the and p16 external benignis nevia granular cyclin-dependent are derived layer of from the kinase neuroectoder- cerebellum. inhibitor, posed47 kDa of consisting 10 exons of extended 324 amino over acids. 10 kbIts of 3.6 genomic kb cDNA DNA. is com- The Since melanomas and benign nevi are derived from neuroectoder- posed of 10 exons extended over 10 kb of genomic DNA. The whichmal cells blocks in the cell skin, cycle we entry aimed by to regulating assess whether the phosphorylation BMI-1 is also Grant sponsor: UBS AG Switzerland. human BMI-1 protein structurally and with respect to its amino stateexpressedmal of cells pRb byin (retinoblastoma melanomathe skin, we cells. aimed protein). to assess p16ink4a whetherinhibits BMI-1 the kinase is also *Correspondence to: Department of Pathology, University Hospital, acidhuman composition BMI-1 protein is identical structurally to the mouseand with BMI-1 respect protein to its and amino was expressed by melanoma cells. Schmelzbergstrasse 12, 8091 Zurich, Switzerland. Fax: 141-44-255-4416. acid composition is identical to the mouse BMI-1 protein and was activeConsequently, sites of the cyclinwe analyzed D-dependent benign kinases nevi, primary Cdk4 and melanomas, Cdk6 and the first member of the Polycomb gene family identified in mam- preventsConsequently, their associations we analyzed with benign cyclins. nevi,7 p16 primaryink4a normally melanomas, is E-mail: [email protected] mals.the first4 Low member BMI-1 of theexpression Polycomb isobserved gene family in almostidentified all in tissues, mam- melanoma metastases and cell lines, and demonstrate frequent Received 29 December 2006; Accepted after revision 28 March 2007 4 downregulatedexpressionmelanoma of metastases the upon stem mitogen cell and markers cell stimulation, lines, BMI-1 and and thus demonstrate nestin. promoting Moreover, frequent cell DOI 10.1002/ijc.22891 withmals. higherLow expression BMI-1 expression in the brain, is observed spinal cord, in almost kidney, all lung tissues, and ink4a cycleexpression progression. of the p16 stem cellknock markers out BMI-1 mice develop and nestin. normally, Moreover, but gonads.with higher3,5 In expression addition, inBMI-1 the brain, is highly spinal expressed cord, kidney, in embryonic lung and we show that high BMI-1 expression is associated with an Published online 27 June 2007 in Wiley InterScience (www.interscience. 3,5 havewe increasedshow that cancer high susceptibility BMI-1 expression and develop is associated multiple with malig- an wiley.com). stemgonads. cells andIn addition, the placenta. BMI-11 is highly expressed in embryonic increased risk of metastatic disease. Besides being of prognostic 1 relevance,increased risk these of data metastatic may provide disease. new Besides insights being in melanoma of prognostic tu- stemBMI-1 cellsis and a the transcriptional placenta. repressor of the Ink4a/Arf locus,6 6 morigenesis.relevance, thesePublication data of may the International provide Union new Against insights Cancer in melanoma tu- whichBMI-1 encodes is a transcriptional 2 separate tumor repressor suppressor of the Ink4a/Arf genes, namely locus, morigenesis. p16whichink4a encodesand p14Arf 2(p19 separateArf in mouse),tumor suppressor which are read genes, in alternat- namely p16ink4a and p14Arf (p19ink4aArf in mouse), which are read in alternat- ing reading frames. p16ink4a is a cyclin-dependent kinase inhibitor, whiching reading blocks frames. cell cycle p16 entryis a by cyclin-dependent regulating thephosphorylation kinase inhibitor, Grant sponsor: UBS AG Switzerland. statewhich of blocks pRb (retinoblastoma cell cycle entry protein). by regulating p16ink4a theinhibits phosphorylation the kinase *GrantCorrespondence sponsor: UBS to: AGDepartment Switzerland. of Pathology, University Hospital, ink4a * 1 activestate of sites pRb of (retinoblastoma the cyclin D-dependent protein). kinases p16 Cdk4inhibits and the Cdk6 kinase and SchmelzbergstrasseCorrespondence 12, to: 8091Department Zurich, Switzerland. of Pathology, Fax: University41-44-255-4416. Hospital, Schmelzbergstrasse 12, 8091 Zurich, Switzerland. Fax: 141-44-255-4416. preventsactive sites their of the associations cyclin D-dependent with cyclins. kinases7 p16 Cdk4ink4a andnormally Cdk6 and is E-mail: [email protected] 7 ink4a E-mail:Received [email protected] 29 December 2006; Accepted after revision 28 March 2007 downregulatedprevents their associations upon mitogen with stimulation, cyclins. p16 thus promotingnormally cell is DOIReceived 10.1002/ijc.22891 29 December 2006; Accepted after revision 28 March 2007 cycledownregulated progression. upon p16ink4a mitogenknock stimulation, out mice develop thus promoting normally, butcell ink4a PublishedDOI 10.1002/ijc.22891 online 27 June 2007 in Wiley InterScience (www.interscience. havecycle increased progression. cancer p16 susceptibilityknock out and mice develop develop multiple normally, malig- but wiley.com).Published online 27 June 2007 in Wiley InterScience (www.interscience. have increased cancer susceptibility and develop multiple malig- wiley.com). Publication of the International Union Against Cancer Publication of the International Union Against Cancer STEM CELLS, BMI-1, P16ink4a AND MELANOMA 1765 Material and methods 20 3 106 cells/ml. Cells were centrifuged for 5 min at 2,000 rpm. Patients After removing the supernatant, 4 drops of plasma were added to the pellet and the solution was mixed. One drop of thrombin was Formalin-fixed, paraffin-embedded tissue of 64 primary cutane- added. After 5 min the coagulated material was encapsulated for ous melanomas and 20 skin nevi (4 junctional, 10 dermoepidermal fixation in formalin 4% and embedded in paraffin. Cell cylinders and 6 dermal) were immunohistochemically analyzed for BMI-1 of 0.6 mm in diameter were punched from the ‘‘cell line’’ tissue expression. The 64 patients with primary cutaneous melanomas 34 block and inserted into a recipient paraffin block by utilizing a were previously reported in a sentinel lymph node (SLN) study. custom-made precision instrument as described earlier. About The patient age ranged from 20 to 75 years. Clinical follow-up l 6 4.0 m sections of the resulting cell line microarray were further was available in 54 of 64 patients (mean clinical follow up, 34 processed as described earlier. An experienced cytologist (SK) 12 month). There were 35 nodular, 13 superficial spreading, 1 len- reviewed all 53 cell lines and confirmed the presence of melanoma tiginous, 7 acral lentiginous, 7 not otherwise specified and 1 des- cells. moplastic malignant melanoma. All melanomas had a Breslow tumor thickness 1.0 mm. Thirty four of the 64 patients had a Breslow tumor thickness from 1 to 2 mm, while tumor thickness Immunohistochemistry was more than 2 mm in 30 patients. Forty three of 64 patients BMI-1 immunohistochemistry was performed on paraffin sec- (67%) had a pN0 (sn; i-) SLN stage category. Nineteen of 64 tions of formalin-fixed tissues, using a Ventana Benchmark auto- patients had a pN1 or pN2 stage after lymphadenectomy. Five mated staining system (Ventana Medical Systems, Tucson, Ari- patients with negative SLN developed clinically obvious lymph zona) as recently described.19 For antigen retrieval, slides were node (n 5 4) or organ (n 5 1) metastases during follow-up. Tu- heated with a cell conditioner 1 (standard procedure). Endogenous mor-matched samples of primary melanoma and metastasis were biotin was blocked with the appropriate kit. Primary monoclonal available in 8 patients. Eight of 54 (15%) patients died of meta- mouse antibody against BMI-1 (Upstate, Lake Placid, NY, clone static disease during clinical follow-up. All except 1 had positive F6, dilution 1:50) was then applied and revealed with the iVIEW SLNs at the time of primary diagnosis. DAB detection kit, yielding a brown reaction product. The signal To screen for BMI-1 expression in melanoma metastases, paraf- was enhanced with the Ventana amplification kit. Slides were fin-embedded tissue of 165 melanoma metastases of different ana- counterstained with hematoxylin prior to glass coverslipping. tomic sites were identified in the archives of the Institute for Sur- Nestin and p14Arf immunohistochemistry was performed as gical Pathology Zurich. They consist of 116 brain, 24 lymph node, described earlier. Primary monoclonal mouse antibodies against 7 soft tissue, 5 gastrointestinal, 4 skin, 4 lung, 2 liver, 1 parotis, 1 nestin (Chemicon, MAB5326, dilution 1:100) and p14Arf (Labvi- bone and 1 adrenal metastases. sion Coporation, RB-9402-P1, dilution 1:25) were used. p16ink4a immunohistochemistry was performed as recently described.37,38 In addition, cell lines derived from 53 melanomas (22 primary ink4a cutaneous melanomas, 26 metastases and 5 of unknown sites) Primary monoclonal mouse antibody against p16 (Neo- were grown. The cell lines from metastases were from 12 skin, 6 markers, Freemont, CA, Clone 16P04, dilution 1:600) was used. lymph node, 3 cerebral and 1 gastric metastases. In 4 patients the origin was unknown. The melanoma cells were obtained from sur- Statistics gical specimen of melanoma patients included in a clinical trial. BMI-1, p16ink4a, p14Arf and nestin expression in primary mela- All patients provided written informed consent on the use of their noma were compared between different patient groups using the tissue samples for the establishment of melanoma cell cultures. Mann-Whitney test. Correlations between BMI-1, p16ink4a, p14Arf, nestin and Breslow tumor thickness were analyzed using Spear- Melanoma cell cultures man’s rank correlation. Differences in tumor-specific survival between groups were calculated by log rank test. A logistic regres- Melanoma cells were released from washed and fragmented tis- sion was performed to evaluate the predictive power of BMI-1 and sue sections by means of serial incubation with dispase and colla- ink4a p16 expression in primary malignant melanoma for lymph genase as previously described.35 Outgrowing melanoma cells node metastasis. p-Values below 0.05 were considered as signifi- were routinely cultivated as monolayers in RPMI-1640 supple- cant. SPSS 12.0.1 for windows (SPSS) was used for statistical mented with 10% heat inactivated fetal calf serum (Seromed, Ber- analyses. lin, Germany), 5 mM glutamine (Biochrom KG, Berlin, Ger- TM 1768 MIHIC-PROBST ET AL. many), 1 mM sodium pyruvate (Gibco Invitrogen AG, Basel, INK4A ° TABLE II – RELATIVE RISK OF LYMPH NODE METASTASIS ACCORDING TO BMI-1 AND P16 EXPRESSION LEVELS IN PRIMARY MELANOMA Switzerland) and 1% antibiotic mixture (Gibco) at 37 C and in Results n Univariate OR p-value Multivariate OR p-value p16ink4a low vs. high1 35/29 3.0 (1.0–8.6)2 0.04 2.7 (0.89–8.1) 0.08 5% CO2 atmosphere. The melanoma cells proliferated in culture BMI-1 high vs. Low1 ink4a Arf41/23 4.5 (1.3–15.6) 0.02 4.1 (1.2–14.6) 0.03 BMI-1,ink4a p16 ,1 p14 and nestin expression in benign nevi with a median growth time of 66 days (range 4–287 days). Cul- p16 low/BMI-1 high vs. others 22/42 3.2 (1.4–7.3) 0.005 OR,and odds ratio; melanomas CI, confidence interval. tures were within 3 passages at processing. Melanoma cell line 1For definition see text.–2Values in parentheses indicate 95% CIs. UKRRV-Mel2 (kindly provided by Prof. Dirk Schadendorf, Ger- Using the primary monoclonal mouse antibody against BMI-1 (clone F6) according to our recently reported staining procedure,19 man Cancer Research Center, Skin Cancer Unit, Heidelberg, Ger-Master of Science in Medical Biology 5 64 (16%)spermatogonia primary melanomas, 32 of and 165 metastases lymphocytes (19%) and 13 showedBMI-1 is highly nuclear expressed in BMI-1 specific neuroepithelial expres- cells. many), freshly isolated human fibroblasts and Jurkat lymphoma of 53 (25%) cell lines were p14Arf positive. All 20 nevi were Therefore, it was not surprising that all benign nevi and 60–70% p14Arfsionnegative. with Neither a nuclear strong nor cytoplasmic staining p14Arf intensity.expres- of melanomas Therefore, showed high BMI-1 spermatogonia expression levels. Melanocytes in cell line (American Type Culture Collection, Manassas, VA) were sion was associated with BMI-1-, p16ink4a-expression or meta- are neuroectodermal cells, and hence, markers of neuroectodermal used as controls. statictestes disease. were used as external positivedifferentiation control. can be found in Lymphocytes almost all melanomas. Importantly, in For nestin, a cytoplasmic immunostaining was detectable. Fifty high BMI-1 expression levels may serve as an argument for stem six ofinvestigated 64 primary melanomas benign were available nevi for nestin and evaluation. melanomascell properties ofserved melanoma as cells. positive BMI-1 is highly inter- expressed in Using the 50% cutoff to define tumors with high nestin expression, both hematopoietic and neural stem cells. In the brain, BMI-1 is 14 ofnal 56 (25%) control. primary melanomas, Almost 84 of all 165 metastasesbenign (50%) neviexpressed and melanomasin the subventricular zone,showed the source BMI-1 of neural stem cells, as well as in cortical neural stem cells and progenitor Tissue microarray construction from metastases and cell lines and 21 of 53 cell lines (40%) showed increased nestin expression 20,49 (Fig.expression 2). There was a significant in single correlation cellsbetween BMI-1 or a and faintcells. nuclearThe expression staining, of the stem cell which marker nestin was is also nestin expression in cell lines (p 5 0.001) and melanoma metasta- consistent with a particular high percentage of melanoma cells Paraffin-embedded tissue of 165 metastases was used for the ses (pregarded5 0.02), but not as in primary unspecific. melanoma. Nestin To expression excludewith stem the cell possibilitycharacteristics in a large of subgroup unspecific of human mela- was not associated with lymph node metastases. noma. Nestin expression showed high intratumoral heterogeneity preparation of a melanoma metastases tissue microarray. A mor- staining and to clearly define melanomaswith positive and negative with melanoma increased cells within BMI-1 the same tumor. There was a significant correlation between BMI-1 and nestin phologically representative region of the paraffin ‘‘donor’’ blocks BMI-1expression, and melanoma progression a high cutoff level wasexpression used in melanoma to define metastases and tumors melanoma cell with lines, but Tumor thickness (p 5 0.002), LN status (p < 0.0001) and not in primary melanoma. Importantly, melanoma metastases are was chosen. The representative region was taken with a core tissue p16ink4aincreasedexpression (p 5 BMI-10.001) were expression. associated with tumor-spe- Highmore BMI-1 frequently positiveexpression for nestin than was for primary defined melanomas, biopsy (diameter, 0.6 mm; height, 3–4mm) and precisely arrayed cificas survival the but same there was stainingno association between intensity BMI-1 and aswhich lymphocytes is in line with a previous in atstudy. least28 It is tempting 50% to speculate of nestin expression and survival. Univariate logistic regression anal- that melanoma cells with stem cell properties have an increased into a new ‘‘recipient’’ paraffin block using a customer-built ysis inthe primary tumor malignant cells.melanoma demonstrated Otherpatterns that high BMI- weremetastatic categorized behavior and those stem as cells low or stem BMI-1cell-like cells are 36 1 and low p16ink4a expression are predictors of metastatic disease therefore enriched in melanoma metastases. instrument. After the block construction was completed, 4.0-lm (p 5expression.0.02 and 0.04; respectively; Table II). To analyze the correla- Specific stem cell populations may account for tumor forma- tion between BMI-1-, p16ink4a-expression and metastatic disease, 21-23,50,51 ink4a tion and stem cell renewal factors like BMI-1 have been sections of the resulting tumor tissue microarray block were cut 3 categories were defined. Category 1: low BMI-1/high p16 implicated in the generation of different tumors. Recent analyses expression;All Category 20 2: benign high BMI-1/high nevi p16ink4a (100%),or low BMI-1/ 41 of 64 (64%) primary cutaneous ink4a ink4a indicate a tumorigenic subpopulation with stem cell properties in for further analysis. lowmelanomas p16 expression; Category and117 3: high of BMI-1/low 165 p16 melanomamelanoma. Twometastases recent studies26,27 (71%)described subpopulations showed with expression. There were 15 melanomas in Category 1, 27 in Cate- stem cell properties in melanomas and melanoma cell lines. These For the cell line array, 2 drops of melanoma cell culture suspen- goryhigh 2 and 22 BMI-1 in Category 3. expression Interestingly, primary (Fig. melanomas 1). in Therecells maintained was stemno cell difference characteristics after in cloning BMI-1 and were Category 3 had a higher probability for metastases than all other cat- ink4a able to self-renew and differentiate into melanogenic, adipogenic, sion was added to a Cytolyt-solution, when the cell count reached egories.expression The statistical significance between of combined primary BMI-1 and p16 melanomaschondrogenic and and osteogenic metastases. lineages.27 Immature Fifteen melanocytic are analysis was even stronger (p 5 0.005) than for BMI-1 alone (p 5 present in normal human epidermis26 and melanocytic stem cells 0.03). In multivariate logistic regression analysis, high BMI-1 have been identified in the murine hair follicle.52 The high risk of expression was an independent predictor of metastatic disease, ink4a metastatic disease of malignant melanoma, often with delayed whereas p16 negativity did not reach statistical significance. onset, is frequently explained by more primitive stem cell biology. Importantly, high BMI-1 expression levels in melanoma were associated with lymph node metastases. Our finding is consistent Discussion with observations of BMI-1 expression in breast cancer: Kim et Here we demonstrate that a significant percentage of human al. showed a positive correlation between BMI-1 overexpression in invasive ductal breast cancer and presence of axillary lymph melanomas has stem cell properties and that these tumors are asso- 44 ciated with higher incidence of metastases. Different molecular node metastases. A prognostic relevance of BMI-1 expression was also shown for non-Hodgkin B-cell lymphomas and the mye- mechanisms are involved in maintenance of the self-renewal 45,48 capacity of stem cells, which also can contribute to tumorigenesis. lodysplastic syndrome. Highly aggressive non-Hodgkin B-cell lymphomas had significantly higher BMI-1 levels compared to The most prominent example is the Polycomb group gene BMI-1, 48 whose role in control of self-renewal of hematopoetic and neural low-grade malignant B-cell non-Hodgkin lymphoma. Recently, a correlation between BMI-1 expression and poor prognosis was stem cells and its overexpression in leukemia and brain tumors, 47 most notably medulloblastomas,19 is well known. BMI-1 protein also described for nasopharyngeal carcinomas. expression was already reported in various human can- Interestingly, Raaphorst et al. reported increasing BMI-1 cers.14,15,18,31,42-48. Using an immunohistochemical approach, we expression levels during progression from ductal carcinoma in situ describe for the first time BMI-1 expression in human malignant to invasive cancer.46 A stepwise progression model with accumu- melanomas. Immunohistochemically, weak nuclear expression lation of genetic alterations has been proposed for breast cancer.53 with low staining intensity was observed in many different human Therefore, neoexpression of BMI-1 in less-differentiated tumors tissues.1 Therefore, we used very stringent criteria to define in- could explain the association with metastases in breast cancer. creased BMI-1 expression levels in melanocytic lesions. Increased This is in contrast to melanoma, because all nevi and the majority BMI-1-expression was given, if at least 50% of melanocytic cells of melanomas are BMI-1 positive. One might hypothesize that were positive with a staining intensity similar to lymphocytes and specific molecular mechanisms are responsible for the increased spermatogonia. metastatic behavior of BMI-1 positive tumors. Odds ratio (OR) Example: Identification of risk factors for lymph node metastases with prostate cancer (Brown, 1980) n = 52 patients y = nodal metastases (0 = none, 1 = metastases) x = age, phosphatase, X-ray result, tumor size, tumor grade. The first two x–variables are continuous, the rest binary.

Contingency table for the relation between nodal metastases and X-ray result X-ray result x = 0 x = 1 no nodal metastases (y = 0) 28 4 32 nodal metastases (y = 1) 9 11 20 37 15 52 sensitivity = 11/20 = 55% , specificity = 28/32 = 87% χ2–test p = 0.001

Master of Science in Medical Biology 6 Relative risk (RR) or odds ratio (OR)? x = 0 x = 1 y = 0 28 4 32 y = 1 9 11 20 37 15 52

“Risk” defined as P(y = 1|x) = p(x) , → p(0) = 9/37 = 24%, p(1) = 11/15 = 73% 11 × 37 RR = p(1)/p(0) = = 3.0 15 × 9 RR only valid for representative sample

From betting we know “odds”: P(y = 1|x) p(x) = P(y = 0|x) 1 − p(x)

Master of Science in Medical Biology 7 Master of Science in Medical Biology 8 Relative risk (RR) or odds ratio (OR)? Im epidemiology the “odds ratio” is a measure for the relative risk:

x = 0 x = 1 y = 0 28 4 @I@  y = 1 9 © @@R 11

P(y = 1|x = 1)  P(y = 1|x = 0) 28 × 11 OR = = = 8.6 1 − P(y = 1|x = 1) 1 − P(y = 1|x = 0) 9 × 4

OR is also valid for case–control studies

For rare diseases, OR and RR are nearly equal:

p(1)  p(0) p(1) OR = ≈ 1 − p(1) 1 − p(0) p(0)

Master of Science in Medical Biology 9 Modelling by means of logistic regression

What is fundamental for a (simple) regression?

Model: yi = f (xi , β) + εi (i = 1,..., n)

where: f = pre-specified function e.g. linear f (xi , β0, β1) = β0 + β1 xi regression function f (x, β) = conditional expectation of y, given the value x, i.e. E(y | x) = f (x, β)

“outcome” binary event: “success” (y = 1), “failure” (y = 0) probability for success p = P(y = 1) E(y) = 0 × P(y = 0) + 1 × P(y = 1) = p

Master of Science in Medical Biology 10 Why not use ordinary regression? Example: y = presence of nodal metastases x = phosphatase (logarithmised)

●● ● ● ● ●● ●● ●●● ● ● ● ● 1.0 0.6 0.2

● ●●●●● ● ●● ● ●● ●● ● ●●● ● ●● ● P(nodal metastases) −0.2 0.3 0.4 0.6 0.8 1 1.5 phosphatase regression = conditional mean of y given x −→ E(y | x). Thus:E( y | x) = P(y = 1 | x) = p(x)

A probability is modelled — lies between 0 and 1. −→ plausible to model p(x) as distribution function.

Master of Science in Medical Biology 11 ●● ● ● ● ●● ●● ●●● ● ● ● ● 0.8 0.4 P(nodal metastases)

● ●●●●● ● ●● ● ●● ●● ● ●●● ● ●● ● 0.0 0.3 0.4 0.6 0.8 1 1.5 phosphatase

2 16 10 4 0 y = 0 @ @  I@ @ I II ©@ © @ ©@ R ©RR R@ y = 1 0 4 8 6 2 OR ∞ 3.2 1.9 ∞ 16 × 8 OR for [0.58–0.79] vs. [0.41–0.57] = 10 × 4 = 3.2 OR for [0.80–1.09] vs. [0.41–0.57] 16 × 6 16 × 8 10 × 6 = 4 × 4 = 10 × 4 × 4 × 8 = 3.2 × 1.9 = 6 OR for a change of more than one class: multiplicative

Master of Science in Medical Biology 12 Which distribution function to use?

Assumption: odds ratio for adjoining classes is constant (similar to the assumption of a constant slope of the regression function in linear regression) As OR multiplicative, log(OR) must be linear. −→ for log–odds (logits):

 p(x)  log = β + β x 1 − p(x) 0 1

(log = natural logarithm = loge )

−→ p(x) is logistic distribution function

exp(β + β x) p(x) = 0 1 1 + exp(β0 + β1x)

Master of Science in Medical Biology 13 ♣ Linearity of the logit–transformation Assumption: OR for x = x0 + c vs x = x0 is constant in x0 = OR(c)

OR multiplicative −→ OR(c) = OR(1)c

 p(x)  Is g(x) = log linear? 1 − p(x)

OR(c): true OR for “x = c” vs x = 0

log (OR(c)) = g(c) − g(0) logarithmise: g(x) − g(0) = log (OR(1)) x g(x) = g(0) + log (OR(1)) x g(x) = β0 + β1 x

with β0 = g(0) and β1 = log (OR(1)) Master of Science in Medical Biology 14 Estimation and testing in logistic regression

A. How to estimate β0, β1? B. How to test whether the influence of x on y is not by chance (“significant”)?

Scientific hypothesis H1: β1 6= 0 Example: phosphatase influences presence of nodal metastases

Null hypothesis H0: β1 = 0 Example: phosphatase has no influence

Master of Science in Medical Biology 15 Method: Maximum Likelihood Estimation

Attractive characteristics: I. maximum likelihood estimates are optimal (−→ optimal use of data). II. they are normally distributed with known variance–covariance matrix. (−→ precision known −→ statistical tests) III. tests and confidence intervals are optimal (“likelihood ratio tests”) But: iterative procedure, i.e. solution not always correct p–values only valid for large n (“asymptotically”) (analogous to χ2–test)

Master of Science in Medical Biology 16 What is maximum likelihood principle? (informal)

Probability for event yi = 1 known (Bernoulli), depending on unknown model parameters β0, β1 (“likelihood–function”) Inserting data in model for p(x) yields likelihood function (= function of parameters β0, β1):

exp(β0 + β1xi ) P(yi = 1|xi ) = 1 + exp(β0 + β1xi )

Determine βˆ0, βˆ1 by maximising the likelihood, i.e. probabilities to observe these data (xi , yi ) get maximal. Computing: iteratively solve a system of non–linear equations for βˆ0, βˆ1

variance–covariance matrix for βˆ0, βˆ1 as a byproduct. ⇒ Leads to confidence intervals and tests

Master of Science in Medical Biology 17 Example: prostate cancer

Estimate Std. Error z value Pr(>|z|) (Intercept) 0.9919 0.6033 1.64 0.1001 log2(phosph) 2.4198 0.8778 2.76 0.0058

95% confidence interval for exp(β1):

exp(Estimate) Lower Upper

log2(phosph) 11.24 2.01 62.83

Hosmer and Lemeshow test (goodness of fit): χ2 = 7.245, df = 8, p–value= 0.510

Master of Science in Medical Biology 18 Wald test Test for a single predictor Wald test statistic βˆ W = 1 ˆ SE(c β1)

p–value: use of approximate normal distribution of βˆ1 and standard error. Example: Nodal metastases vs. phosphatase Estimate Std. Error z value Pr(>|z|) (Intercept) 0.9919 0.6033 1.64 0.1001 log2(phosph) 2.4198 0.8778 2.76 0.0058 βˆ 2.42 W = 1 = = 2.8 ˆ SE(c β1) 0.9 Two-sided approximate p–value: P(|z| > 2.8) = 0.006 Statistically significant, clinically negative influence of an increased phosphatase Master of Science in Medical Biology 19 Interpretation of coefficients

Linear regression: If x changes by one unit, the mean of y changes by β1 units.

Relation between p(x) = P(y = 1 | x) and x is linear in logits:

 p(x)  g(x) = log = β + β x 1 − p(x) 0 1

Thus: change in x by one unit

−→ change in logit of p(x) by β1 units

Master of Science in Medical Biology 20 Interpretation: binary x variable

“odds ratio” OR: ratio of odds for x = 1 (pos X-ray) to odds for x = 0 (neg X-ray)

p(1)  p(0) OR = 1 − p(1) 1 − p(0) −→ log(OR) = g(1) − g(0)

= (β0 + β1 × 1) − (β0 + β1 × 0)

= β1

i.e. OR = exp(β1)

OR for neg vs. pos X-ray = exp(−β1) = 1/ exp(β1)

Master of Science in Medical Biology 21 Interpretation: continuous x variable

If x changes by one unit, the logit changes by log(OR) = β1 units.

Thus: odds ratio = exp(β1) is a measure for an increase in risk (in odds) when x changes by one unit.

logit-increase when x changes by k units:

log(OR) = (β0 + β1 × (x + k)) − (β0 + β1 × x) = k × β1

OR for change of x by k units:

k k exp(k β1) = (exp(β1)) = OR

Master of Science in Medical Biology 22 Interpretation of coefficients

Example: OR when phosphatase changes by a factor of 2:

OR = exp(β1) = 11.2

OR for a change by a factor of 1.5:

1.5 = 20.585 −→ OR = 11.20.585 = 4.1

Interpretation: categorical or ordinal x variable One has to introduce binary “design variables”, then interpretation as for binary variables.

Master of Science in Medical Biology 23 Computation of individual risk Representative sample → p(x) and RR appropriate

Example: y = nodal metastases, x = log2(phosphatase) exp(βˆ + βˆ x) absolute individual risk: p(x) = 0 1 1 + exp(βˆ0 + βˆ1x)

RR for patients with one unit increase of log2(phosphatase) compared to meanx ¯ (i.e. doubling of phosphatase): x¯ = −0.63 (corresponds to phosphatase of 2−0.63 = 0.64) exp(1.0 + 2.42 × (−0.63)) p(¯x) = = 0.37 1 + exp(1.0 + 2.42 × (−0.63)) exp(1.0 + 2.42 × 0.37) p(¯x + 1) = = 0.87 1 + exp(1.0 + 2.42 × 0.37) p(¯x + 1) 0.87 −→ RR = = = 2.3 p(¯x) 0.37 individual risk with doubled phosphatase is increased by a factor of 2.3. The OR, however, is OR = 11.2 ! Master of Science in Medical Biology 24 Multiple logistic regression

k > 1 variables x1,..., xk → multiple logistic regression

Reasons as for multiple linear regression: 1 Eliminate potential effects of “confounding” variables in a study with one explanatory variable. 2 Investigate potential prognostic factors of which we are not sure whether they are important or redundant. 3 Develop formulas for a better prediction of individual risk based on explanatory variables

Problem solved with maximum likelihood principle Rule of thumb: at least 20 events and 20 non-events per explanatory variable

Master of Science in Medical Biology 25 Univariate analysis for prostate cancer example

Estimate Std. Error z value Pr(>|z|) OR

log2(phosph) 2.4198 0.8778 2.76 0.0058 11.2 Age -0.0448 0.0468 -0.96 0.3379 1.0 X-ray 2.1466 0.6984 3.07 0.0021 8.6 Size 1.6094 0.6325 2.54 0.0109 5.0 Grade 1.1389 0.5972 1.91 0.0565 3.1

Master of Science in Medical Biology 26 Multiple logistic regresstion: prostate cancer example

Estimate Std. Error z value Pr(>|z|) OR (Intercept) -0.5418 0.8298 -0.65 0.5138 log2(phosph) 2.3645 1.0267 2.30 0.0213 10.6 X-ray 1.9704 0.8207 2.40 0.0163 7.2 Size 1.6175 0.7534 2.15 0.0318 5.0

Interpretation:

βˆi Influence of xi when remaining variables are fixed

p–values Does xi , given the fixed remaining variables, yield additional information about P(y = 1)? Significant variables are called “independent risk factors”.

exp(βˆi ) OR with fixed remaining variables, i.e. OR of a patient with X-ray = 1, Size = 1, by a factor of 2 decreased phosphatase against a patient with X-ray = 0, Size = 0: OR = 5.0 × 7.2/10.6 = 3.4

Master of Science in Medical Biology 27 Multiple logistic regression

How to combine the information of several significant explanatory variables?

PI = βˆ1x1 + βˆ2x2 + ... + βˆk xk is a prognostic index (score).

If PI large ( > cut-point), we predict “y = 1”.

Master of Science in Medical Biology 28 Model choice and model tests

difficult topic −→ expert similar to linear regression

By means of statistical tests (comparison of models) R2 often provided, but use is controversial judge quality of a model by means of sensitivity and specificity −→ “ROC analysis”

Master of Science in Medical Biology 29 Goodness of prediction Example: Nodal metastases with prostate cancer ROC (receiver operating characteristic) curve:

1.0

0.8

0.6

sensitivity 0.4

PI / AUC = 0.87 0.2 Age / AUC = 0.57 X−ray / AUC = 0.71 Size / AUC = 0.69 0.0 Phosphatase / AUC = 0.75

0.0 0.2 0.4 0.6 0.8 1.0

1 − specificity

PI = 2.4 × log2 (phosphatase) + 2 × X-ray + 1.6 × Size area of 0.5 corresponds to complete ignorance.

Master of Science in Medical Biology 30 Choice of variables

Too many: varying parameter, large SEs Too few: outcome not well explained, bias −→ include all variables that are, a priori, of medical interest −→ “principle of parsimony” −→ if: clear idea −→ comparative testing of models

If unclear and many predictors: (i) Compute univariate model for each x variable, eliminate e. g. those with p > 0.2 (ii) Build a multiple model with the remaining variables; eliminate clearly non–significant variables Alternative: stepwise selection

Master of Science in Medical Biology 31 Model building Linearity of logits in x test against nonlinear alternatives (quadratic, Box–Tidwell–test: interaction x ∗ log(x)) transformation of x to linear relation

Interactions Example: y = occurrence of a coronary heart disease x1 = age, x2 = gender Model without interaction (“additive”):

g(x) = β0 + β1x1 + β2x2 Meaning: gender related differences are not depending on age. If gender related differences increase or decrease with age (“specific effect”) −→ modelling including interaction.

g(x1, x2, x3) = β0 + β1x1 + β2x2 + β12x1x2 OR for gender is then depending on age. Master of Science in Medical Biology 32 Literature

Matthews, D. E. and Farewell, V. T. (1988). Using and understanding medical statistics. 2nd ed., Karger. - contrary to many other introductions, this book includes logistic regression and survival analysis, 200 pages.

Hosmer, D. W. and Lemeshow, S. (2000). Applied logistic regression. 2nd ed., Wiley. - explain logistic regression using 6 worked out, medical examples, 373 pages.

Ryan, T. P. (1997). Modern regression methods. Wiley. - Chapter 1–8: Linear regression - Chapter 9: Logistic regression, 59 pages. - Chapter 10–15: Non-parametric, robust, Ridge-, non-linear regression, experimental design Contains exercises and further literature. Theoretically demanding, 515 pages.

Master of Science in Medical Biology 33