Mean and standard deviation

Supplemental Digital Content 2 – Conversions and Methodology

Mean and standard deviation

Mean is defined as follows:

´ x= ∑

i=1 n

x

_i

n

There are n number of observations (usually patients in health economics applications).

x

_i is the attribute of interest (e.g., visual acuity gains of patient

i

).

Variance is defined as follows:

σ

²

= ∑

i=1

n

( ^x

i

− ´ x )

²

n

Standard deviation (SD) is

σ = √ ^σ

² . See page 81 of Hogg and Tanis [1] for a discussion on mean and variance using empirical distribution.

Subgroup means to aggregate mean

Observations belong to k number of mutually exclusive and collectively exhaustive

subgroups

S

₁

, S

₂

, … , S

_k . The aggregation of subgroup means to aggregate mean uses the following formula:

´ x= n

₁

´ x

₁

+n

₂

´ x

₂

+…+ n

_k

´ x

_k

n

_j is the number of observations for subgroup

n j

and

´ x

_j is the subgroup mean for subgroup

j

.

Proof:

n

₁

´ x

₁

+ n

₂

x ´

₂

+…+ n

_k

x ´

_k

n =

n

₁

∑

i∈S1

x

_i

n

₁

+n

₂

∑

i∈S2

x

_i

n

₂

+ …+ n

_k

∑

i∈Sk

x

_i

n

_k

n

¿

i

∑

∈S1

x

_i

+ ∑

i∈S2

x

_i

+ …+ ∑

i∈S_k

x

_i

n

¿

∑

i=1 n

x

_i

n

¿

´ x

Subgroup SD to aggregate SD

σ

²

= n

₁

(σ

₁²

+ ´ x

₁²

)+n

₂

( σ

₂²

+ ´ x

₂²

)+…+n

_k

(σ

_k²

+ ´ x

_k²

)

n −´ x

²

Proof:

Note that

∑

i=1 n

x

_i²

n =

∑

i=1 n

( ^x

i

−´ x )

²

^{+2 x}

i

x ´ −´ x

²

n

¿

∑

i=1 n

( ^x

i

−´ x )

²

n +

∑

i=1 n

2 x

_i

´ x

n −

∑

i=1 n

´ x

²

n

¿

σ

²

+2 ´ x

∑

i=1 n

x

_i

n − n ´ x

²

n

¿

σ

²

+2 ´ x

²

−´ x

² ¿

σ

²

+ ´ x

²

Consequently,

n ( σ

²

+ ´ x

²

) = n

∑

i=1 n

x

_i²

n

¿

∑

i=1 n

x

_i² ¿

∑

i∈S1

x

_i²

+ ∑

i∈S2

x

_i²

+ …+ ∑

i∈Sk

x

_i²

¿

n

₁

( ^σ

12

+ ´ x

₁²

) ⁺ⁿ

2

( ^σ

22

+ ´ x

₂²

) ^{+ …+ n}

k

( ^σ

k2

+ ´ x

_k²

)

Arrange the above expression in terms of σ^{2 .}

σ

²

= n

₁

( ^σ

12

+ ´ x

₁²

) ^{+ n}

2

( ^σ

22

+ ´ x

₂²

) ^{+…+ n}

k

( ^σ

2k

+ ´ x

_k²

)

n −´ x

²

σ = √ ⁿ

¹

^{( σ}

¹²

^{+ ´ x}

¹²

^{) + n}

²

^{( σ}

²²

^{+ ´ n x}

²²

^{) +…+ n}

^k

^{( σ}

^2k

^{+ ´ x}

^k2

^{) − ´ x}

²

logMAR and Early Treatment Diabetic Retinopathy Study (ETDRS) score

Mean logMAR score to mean ETDRS score

The formula was derived is based on data described in Beck et al. [2], which showed an obvious linear relationship between logMAR score and ETDRS score.

y= − x−1.7 0.02

For example, logMAR value of 0.9 is

−0.9−1.7

0.02

=40 in ETDRS score.

SD of logMAR to SD of ETDRS letters

σ

²_ETDRS

= ( 0.02 ⁻¹ )

²

^σ

_logMAR²

↔ σ

_ETDRS

=50 ×σ

_logMAR

Proof:

VAR (aX + b)= a

²

× VAR ( X)

^{, where}

X

is a random variable and

a

and

b

are constants. See page 80 of Hogg and Tanis [1] for the derivation.

Apply this equation on the linear transformation of logMAR score to ETDRS score:

σ

²_ETDRS

=VAR ( ^{− x−1.7} 0.02 )

^¿

^VAR ( 0.02 ⁻¹ × x+ 1.7

0.02 )

^¿

( 0.02 ⁻¹ )

²

^{VAR ( x )}

^¿

( 0.02 ⁻¹ )

²

^σ

^logMAR2

Take square root on both side of the above equation.

σ_ETDRS=50× σ_logMAR

Change in mean of logMAR score to change in mean of ETDRS score y= − x

0.02

Taking difference at the beginning and the end of observation period deletes the constant.

y=y_beginning−y_end ¿−

x

_beginning

−1.7

0.02 + x

_end

−1.7

0.02

¿−

x

_beginning

− x

_end

0.02

¿−

x

0.02

The variance of the change in mean logMAR score can be calculated using the same formula for the variance of mean logMAR score because deleting the constant (

−1.7

0.02

¿ has no effect on variance (

VAR ( aX + b)=VAR(aX )

).

Decimals to ETDRS letters y=

ln ⁡ ( x 0.0199 ) 0.0461

The above formula was derived based on the conversion table in Elliott [3].

Standard error to standard deviation σ

²

=n× s . e .

²

↔ σ = √ ^{n ×s . e .}

²

See Barde and Barde [4] for the definition.

95% confidence interval to standard error ( upper limit− lowerlimit

s . e .=¿ ¿ 3.92

See section 7.7.7.2 of Higgins and Green [5].

Higgins and Green [5] did not provide a justification for the above equation. According to the definition of confidence interval, the following must hold.

P

(

lower limit ≤ μ ≤upper limit

)

=95 %

Confidence interval is typically estimated as follows:

P ( ´ x −1.96 × s . e .≤ μ ≤ x ´ +1.96 × s. e . ) ≈ 95 % ,

Consequently,

upper limit −lower limit=( ´ x + 1.96× s . e . )−( ´ x−1.96 × s . e .)

↔upper limit−lower limit=

(

x− ´´ x

)

+

(

1.96× s . e .+1.96× s .e .

)

↔upper limit−lower limit=3.92× s . e . (upper limit−lowerlimit

↔ s . e .=¿ ¿ 3.92

The p-value to the t-value

t=tinv

(

p , df

)

,

tinv

: the inverse of the t-distribution df : degrees of freedom

p

: the p-value

See section 7.7.3.3 of Higgins and Green [5]. This is a straightforward application of an inverse function.

Coefficient estimate and t-value to standard error s . e .= ^ β

t ,

^ β

: coefficient estimate

t

: the t-value

See section 7.7.3.3 of Higgins and Green [5].

Again, Higgins and Green [5] did not provide a justification for the above equation. The definition of the t-value is as follows.

t = β ^ − β s . e .

β

is a population parameter, which is not random. Typically, null hypothesis is

β=0

. For example, one can be interested whether the mean difference in visual acuity at baseline and at

year 2 is zero or not. In this case, the expression is reduced to

t = ^ β

s . e .

. A simple rearrangement leads to

s . e .= ^ β

t

^.

References

1. Hogg RV, Tanis EA. Probability and statistical inference. 7th ed. Pearson. 2005.

2. Beck RW, Moke PS, Turpin AH, et al. A computerized method of visual acuity testing:

adaptation of the early treatment of diabetic retinopathy study testing protocol. Am J Ophthalmol.

2003;135(2):194–205. doi:10.1016/S0002-9394(02)01825-1

3. Elliott DB. The good (logMAR), the bad (Snellen) and the ugly (BCVA, number of letters read) of visual acuity measurement. Ophthalmic Physiol Opt. 2016;36(4):355–58.

doi:10.1111/opo.12310

4. Barde MP, Barde PJ. What to use to express the variability of data: standard deviation or standard error of mean? Perspect Clin Res. 2012;3(3):113–16. doi:10.4103/2229-3485.100662 5. Higgins JP, Green S. Cochrane handbook for systematic reviews of interventions. Vol 4.

Chichester, UK: John Wiley & Sons; 2011.