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We discuss the role that the null hypothesis should play in the construction of a test statistic used to make a decision about that hypothesis. To construct the test statistic for a point null hypothesis about a binomial proportion, a common recommendation is to act as if the null hypothesis is true. We argue that, on the surface, the one-sample t -test of a point null hypothesis about a Gaussian population mean does not appear to follow the recommendation. We show how simple algebraic manipulations of the usual t-statistic lead to an equivalent test procedure consistent with the recommendation. We provide geometric intuition regarding this equivalence and we consider extensions to testing nested hypotheses in Gaussian linear models. We discuss an application to graphical residual diagnostics where the form of the test statistic makes a practical difference. By examining the formulation of the test statistic from multiple perspectives in this familiar example, we provide simple, concrete illustrations of some important issues that can guide the formulation of effective solutions to more complex statistical problems., Statistica, Vol. 82 No. 2 (2022)

Circulating insulin-like growth factor-1 (IGF-1) is consistently associated with prostate cancer risk. IGF-1 binds to IGF-1 receptor (IGF1R) and insulin receptor (IR), activating cancer hallmark pathways. Experimental evidence suggests that TMPRSS2:ERG may interact with IGF/insulin signaling to influence progression. We investigated IGF1R and IR expression and its association with lethal prostate cancer among 769 men. Protein expression of IGF1R, IR and ERG (i.e. a surrogate of ERG fusion genes) were assayed by immunohistochemistry. Cox models estimated hazard ratios (HR) and 95% confidence intervals (CI) adjusted for clinical characteristics. Among patients, 29% had strong tumor IGF1R expression and 10% had strong IR expression. During a mean follow-up of 13.2 years through 2012, 80 men (11%) developed lethal disease. Tumors with strong IGF1R or IR expression showed increased cell proliferation, decreased apoptosis and a higher prevalence of ERG. In multivariable models, strong IGF1R was associated with a borderline increased risk of lethal prostate cancer (HR 1.7; 95% CI 0.9-3.1). The association appeared greater in ERG-positive tumors (HR 2.8; 95% CI 0.9-8.4) than in ERG-negative tumors (HR 1.3; 95% CI 0.6-3.0, p-heterogeneity 0.08). There was no association between IR and lethal prostate cancer (HR 0.8; 95% CI 0.4-1.9). These results suggest that tumor IGF1R expression may play a role in prostate cancer progression to a lethal phenotype and that ERG-positive tumors may be more sensitive to IGF signaling. These data may improve our understanding of IGF signaling in prostate cancer and suggest therapeutic options for disease subtypes.

Integrating genomic information with traditional clinical risk factors to improve the prediction of disease outcomes could profoundly change the practice of medicine. However, the large number of potential markers and possible complexity of the relationship between markers and disease make it difficult to construct accurate risk prediction models. Standard approaches for identifying important markers often rely on marginal associations or linearity assumptions and may not capture non-linear or interactive effects. In recent years, much work has been done to group genes into pathways and networks. Integrating such biological knowledge into statistical learning could potentially improve model interpretability and reliability. One effective approach is to employ a kernel machine (KM) framework, which can capture nonlinear effects if nonlinear kernels are used (Scholkopf and Smola, 2002; Liu et al., 2007, 2008). For survival outcomes, KM regression modeling and testing procedures have been derived under a proportional hazards (PH) assumption (Li and Luan, 2003; Cai, Tonini, and Lin, 2011). In this article, we derive testing and prediction methods for KM regression under the accelerated failure time (AFT) model, a useful alternative to the PH model. We approximate the null distribution of our test statistic using resampling procedures. When multiple kernels are of potential interest, it may be unclear in advance which kernel to use for testing and estimation. We propose a robust Omnibus Test that combines information across kernels, and an approach for selecting the best kernel for estimation. The methods are illustrated with an application in breast cancer.

Loss of PTEN has been shown to be associated with aggressive behavior of prostate cancer. It is less clear that loss of PTEN also increases the risk of cancer mortality. We investigated the association between PTEN expression and prostate cancer mortality and the potential effect modification by IGF-IR, a direct activator of the phosphoinositide-3-kinase (PI3K) pathway.Protein expression in tumor was evaluated using tumor tissues obtained from 805 participants of the Physicians' Health and the Health Professionals Follow-up studies who were diagnosed with prostate cancer and underwent radical prostatectomy. Proportional hazard models were used to assess PTEN expression and its interaction with IGF-IR, in relation to lethal prostate cancer (cancer-specific death or distant metastases).Low PTEN expression was associated with an increased risk of lethal prostate cancer [HR, 1.7; 95% confidence interval (CI), 0.98-3.2; Ptrend = 0.04]. The association was attenuated after adjustment for Gleason grade, tumor stage, and prostate-specific antigen (PSA) at diagnosis. A significant negative interaction between PTEN and IGF-IR was found (Pinteraction = 0.03). Either reduction in PTEN or increase in IGF-IR expression was sufficient to worsen prognosis. Models including PTEN and IGF-IR expression offer additional predicting power to prostate cancer survival, compared to those only including demographic and clinical factors.Low PTEN protein expression significantly increases the risk of lethal prostate cancer, particularly when the IGF-IR expression remains at normal level.PTEN and IGF-IR expression in tumor are promising candidates for independent prognostic factors to predict lethal prostate cancer.

The number of available genes from the seven selected KEGG-defined metabolic pathways and the number of common genes between the pathways, after the exclusion of genes that failed quality control. Table S2 Genes associated with tumorigenesis in 202 prostate cancer cases with tumor and normal tissue available according to a conditional logistic regression model and a Least Absolute Shrinkage and Selection Operator (LASSO) regression model. Table S3 Genes associated with Gleason grade (Grade 2–7 versus grade 8) in 404 prostate cancer cases according to a logistic regression model and a Least Absolute Shrinkage and Selection Operator (LASSO) regression model. Table S4 Genes associated with lethal disease in 404 prostate cancer cases (291 non-lethal and 113 lethal) cases according to a logistic regression model and a Least Absolute Shrinkage and Selection Operator (LASSO) regression model. Table S5 Global test p value for pathway level associations with lethal prostate cancer exploring potential effect modifiers. Figure S1 Strength (log odds ratios) and significance (log p values) of individual gene associations with Gleason grade ≥8 (n = 106) versus Gleason grade 2–7 (n = 298) tumors across the seven pathways. Figure S2 Strength (log odds ratios) and significance (log p values) of individual gene associations with lethal prostate cancer among the seven metabolic pathways after additional adjustment for Gleason grade. (DOCX 337 kb)

The number of available genes from the seven selected KEGG-defined metabolic pathways and the number of common genes between the pathways, after the exclusion of genes that failed quality control. Table S2 Genes associated with tumorigenesis in 202 prostate cancer cases with tumor and normal tissue available according to a conditional logistic regression model and a Least Absolute Shrinkage and Selection Operator (LASSO) regression model. Table S3 Genes associated with Gleason grade (Grade 2–7 versus grade 8) in 404 prostate cancer cases according to a logistic regression model and a Least Absolute Shrinkage and Selection Operator (LASSO) regression model. Table S4 Genes associated with lethal disease in 404 prostate cancer cases (291 non-lethal and 113 lethal) cases according to a logistic regression model and a Least Absolute Shrinkage and Selection Operator (LASSO) regression model. Table S5 Global test p value for pathway level associations with lethal prostate cancer exploring potential effect modifiers. Figure S1 Strength (log odds ratios) and significance (log p values) of individual gene associations with Gleason grade ≥8 (n = 106) versus Gleason grade 2–7 (n = 298) tumors across the seven pathways. Figure S2 Strength (log odds ratios) and significance (log p values) of individual gene associations with lethal prostate cancer among the seven metabolic pathways after additional adjustment for Gleason grade. (DOCX 337 kb)

Circulating insulin-like growth factor-1 (IGF-1) is consistently associated with prostate cancer risk. IGF-1 binds to IGF-1 receptor (IGF1R) and insulin receptor (IR), activating cancer hallmark pathways. Experimental evidence suggests that TMPRSS2:ERG may interact with IGF/insulin signaling to influence progression. We investigated IGF1R and IR expression and its association with lethal prostate cancer among 769 men. Protein expression of IGF1R, IR and ERG (i.e. a surrogate of ERG fusion genes) were assayed by immunohistochemistry. Cox models estimated hazard ratios (HR) and 95% confidence intervals (CI) adjusted for clinical characteristics. Among patients, 29% had strong tumor IGF1R expression and 10% had strong IR expression. During a mean follow-up of 13.2 years through 2012, 80 men (11%) developed lethal disease. Tumors with strong IGF1R or IR expression showed increased cell proliferation, decreased apoptosis and a higher prevalence of ERG. In multivariable models, strong IGF1R was associated with a borderline increased risk of lethal prostate cancer (HR 1.7; 95% CI 0.9-3.1). The association appeared greater in ERG-positive tumors (HR 2.8; 95% CI 0.9-8.4) than in ERG-negative tumors (HR 1.3; 95% CI 0.6-3.0, p-heterogeneity 0.08). There was no association between IR and lethal prostate cancer (HR 0.8; 95% CI 0.4-1.9). These results suggest that tumor IGF1R expression may play a role in prostate cancer progression to a lethal phenotype and that ERG-positive tumors may be more sensitive to IGF signaling. These data may improve our understanding of IGF signaling in prostate cancer and suggest therapeutic options for disease subtypes.