Table 2.

Summary of the conditions needed for the generalized and modified basic network scale-up estimators, and their components, to produce estimates that are consistent and essentially unbiased. This table uses the version of the basic scale-up estimator we recommend in Section 4.2.

Quantity	Conditions required	Condition type	Result
reported connections to H (ŷF,H)	1. probability sample from F	sampling	B.1
average personal network size of F ( ${\widehat{\overline{d}}}_{F,F}$)	1. probability sample from F	sampling	B.3
	2. groups of known size total is accurate N𝒜	survey construction
	3. probe alter condition (d̄𝒜,F = d̄F,F )	survey construction
	4. accurate reporting condition (yF,𝒜 = dF,𝒜)	reporting behavior
average visibility of H ( ${\widehat{\overline{v}}}_{H,F}$)	1. relative probability sample from H	sampling	C.2
	2. groups of known size total is accurate N𝒜H∩F	survey construction
	3. probe alter condition ( $\frac{v_{H,\mathcal{A}\cap F}}{N_{\mathcal{A}\cap F}}=\frac{v_{H,F}}{N_F}$)	survey construction
	4. accurate aggregate reports about visibility (ṽH,𝒜H∩F = vH,𝒜H∩F )	reporting behavior
generalized scale-up ( ${\widehat{N}}_H=\frac{{\widehat{y}}_{F,H}}{{\widehat{\overline{v}}}_{H,F}}$)	1. conditions needed for ŷF,H	sampling	C.8
	2. conditions needed for ${\widehat{\overline{v}}}_{H,F}$	sampling, survey construction, reporting behavior
	3. no false positive reports about connections to H (ηF = 1)	reporting behavior
modified basic scale-up ( ${\widehat{N}}_H=\frac{{\widehat{y}}_{F,H}}{{\widehat{\overline{d}}}_{F,F}}$)	1. conditions needed for ŷF,H	sampling	Sections 2–3
	2. condition needed for ${\widehat{\overline{d}}}_{F,F}$	sampling, survey construction, reporting behavior
	3. no false positive reports about connections to H (ηF = 1)	reporting behavior
	4. members of H and members of F have same average personal network size (δF = 1)	network structure
	5. no false negative reports about connections to H (τF = 1)	reporting behavior