(Arminius is the Latin form of the German given name, Hermann.)
Where was Joseph in the 1860 census?  Is the Joseph STROUP who married Eliza ROUSH really their son?
And where was Matilda in the 1850 census?  Is she really their daughter?

1a.  Anon.  1966.  Marriage Index:  Ohio, 1789-1850.  Liahona Research, Orem, UT (Broderbund CD-400):

Stroup, Claibourn

Stroup, Mary

22 Dec 1831

Highland Co.

1b.  Jordan R. Dodd, ed.  1998.  Marriage Index: Ohio, 1851-1900.  Liahona Research, Orem, UT (Broderbund CD-236):

Stroup, Claiborne

Perrin, Sarah

25 Sep 1858

Highland Co.

1c.  Marriage Record No. 1175, Highland County, Ohio:  Claburn Stroup (widower) m. Ann B. Moore (widow), also given Anna B. Moor.  License date 27 Aug 1883; ceremony date 28 Aug 1883; married by Isaac Shaffer, J.P. (scan of record courtesy of Stephen Deutschmann).

2.  1840 Census Index and Digital Images (online at Ancestry.com; Image #7-8 of 10; extracted by Diana Gale Matthiesen):¤•

1840

OH

Highland Co.

Dodson Twp.

pp. 102B-103A

Ln. 24

Claiborn Stroup

200 010 - 110 010

0100000

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
2 males	4 or under	1835-1840	= Anthony (b. 1838) = Arminius (b. 1834)
1 male	20-29	1810-1820	= Claiborne (b. 1810)
1 female	4 or under	1835-1840	= Margaret (b. 1836)
1 female	5-9	1830-1835	= Louisa (b. 1832)
1 female	20-29	1810-1820	= Mary (b. 1812)
1	persons engaged in agriculture

3.  1850 Census Index/Images (online at Ancestry.com; Image #30 of 31):  Dodson Twp., Dist. 14, Highland Co., OH, p. 280A/559, 997/997, enumerated 21 Aug 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):¤•

Clayburn Stroup	39	M		Farmer	3300	O
Polly       "	38	F				"
Louisa      "	18	F				"
Armenious   "	16	F				"
Margaret    "	13	F				"
Anthony Jr  "	10	M				"
Mary A      "	7	F				"
Joseph      "	5	M				"
Wyatt       "	1/12	M				"

Where is Matilda?  Was she missed by the enumerator?  Is she really their daughter?

4.  1860 Census Images (online at Ancestry.com; Image #19 of 38):  Dodsonville P.O., Dodson Twp., Highland Co., OH, p. 67A, PN 134, 972/957, enumerated 9 Jun 1860, official enumeration date 1 Jun 1860 (extracted by Diana Gale Matthiesen):¤•

Claybourn Stroup	49	M		Farmer	5130	550	Ohio
Sarah        "	47	F					Del
Anthony      "	21	M					Ohio
Mary Ann     "	19	F					"
Matilda      "	11	F					"
Franklin     "	7	M					"
Lavina       "	14	F					"

Where is Joseph?  And where was Matilda in 1850?  If Matilda is not their daughter, surely she would have been listed last, after Lavina.  Lavina STROUP is Lavinia YOUNG, Sarah's granddaughter from a prior marriage.

5.  1870 Census Images (online at Ancestry.com; Image #7 of 29):  Hillsborough P.O., Dodson Twp., Highland Co., OH, p. 65A, PN 22, 183/174, enumerated 14 Jul 1870, official enumeration date 1 Jun 1870 (extracted by Diana Gale Matthiesen):¤•

Stroup Clayborne	60	M	W	Farmer	5000	1500	Ohio
______ Sarah	59	F	W	Keeping house			Delaware
______ Franklin	17	M	W	Works on farm			Ohio
Moore Ann	36	F	W	Dom Servant			Ohio

Ann (PERRIN) MOORE, is the ex-wife of William D. MOORE and daughter of Sarah from a prior marriage.  Listed next to Claiborne's first cousin, James STROUP. 

6.  1880 Census (indexed at FamilySearch.org; page image online at Ancestry.com, Image #23 of 40):  Dodson Twp., Highland Co., OH, p. 282C, PN 23, SD 4, ED 41, enumerated 12-14 Jun 1880, official enumeration date 1 Jun 1880 (extracted by Diana Gale Matthiesen):¤•

1880:  for an explanation of the column headings, please see What the Numbers in the Federal Census Mean (missing columns contained no data).

1

2

3

4

5

6

8

9

10

11

13

21

22

23

24

25

26

81

88

Stroup Claburne

W

M

70

/

Farmer

OH

PA

PA

______ Sarah

W

F

69

Wife

/

House Keeping

DE

DE

DE

Moore Ann B

W

F

46

Servant

/

House work

/

/

OH

OH

OH

_____ William

W

F

18

/

works on farm

/

OH

OH

OH

Ann Barbara (PERRIN) MOORE is Sarah's daughter, and William MOORE is Ann's son.  Listed next to Claiborne's first cousin, James STROUP, on one side, and to Claiborne's son, Frank P. STROUP, on the other side. 

7.  1890 Census:  the 1890 Census Population Schedules were destroyed.

8.  LDS.  Family Search: Internet Genealogy Service:  AF - Ancestral File (online at FamilySearch.org).

9.  Emails from Stephen Deutschman.